AP Physics C: Electricity and Magnetism FRQ Room

Analysis of Linear Charge Distribution in a Rod

A uniformly charged rod is oriented horizontally. The electric field is measured along the perpendic

Calculating Work Done by an Electric Field in Moving a Particle

In this problem you will calculate the work done by an electric field on a charged particle moving b

Calculus-based Potential Difference from a Point Charge

The electric potential due to a point charge is a function of distance. Analyze and calculate the po

Calibrated Probe Measurements of Electric Field Strength in a 2D Plane

A set of calibrated probes measures the electric field at various (x, y) positions within a planar r

Charge Conservation and Electrostatic Interactions

Charge conservation and electrostatic interactions are fundamental concepts in electromagnetism. Dis

Charge Distribution in a Non-uniformly Charged Object

An experiment is proposed to study the charge distribution on a non-uniformly charged spherical obje

Computation of Electric Flux Through a Curved Surface

A curved surface is placed in a uniform electric field. The cumulative electric flux through the sur

Conductors in Electrostatic Equilibrium

A hollow metallic (conducting) sphere is uniformly charged with a total charge $$Q$$ and contains an

Data Analysis of Electric Force Measurements

In a laboratory experiment, the force between charged spheres was measured at various distances. Usi

Deriving the Potential Difference across a Parallel Plate Capacitor

This problem involves using the definition of electric potential to derive the potential difference

Determining Surface Charge Density on a Conductor

An experiment is conducted to determine the surface charge density on a conductor by measuring the e

Dipole in a Non-Uniform Electric Field: Torque and Equilibrium

A dipole with moment p is placed in a non-uniform electric field given by $$E(x)= E₀*(1+β*x)$$. Usin

Electric Field Along the Axis of a Charged Ring

A thin ring of radius $$R$$ carries a uniform charge Q distributed evenly along its circumference. D

Electric Field Due to a Uniformly Charged Sphere

In this problem you will derive the electric field expressions for points both outside and inside a

Electric Field from a Uniformly Charged Rod

A researcher is studying the electric field produced by a thin rod of length $$L$$, uniformly charge

Electric Field Mapping and Superposition

This problem focuses on constructing electric field lines for multiple charge configurations and app

Electric Field Mapping with a Charged Sphere

An experimenter seeks to measure the electric field produced by a uniformly charged metallic sphere.

Electric Field of a Charged Ring: A Calculus Approach

A ring of radius $$R$$ carries a uniform charge distribution with total charge $$Q$$. Determine the

Electric Field of a Conducting Sphere

A conducting sphere of radius R is uniformly charged with total charge Q. Answer the following quest

Electric Field Using Gauss' Law for a Cylindrical Charge Distribution

An infinitely long cylinder carries a uniform charge per unit length. Determine the electric field b

Electric Flux through a Non-spherical Surface

An electric field originating from a point charge is present and you are to evaluate the electric fl

Electric Potential and Field Inside a Dielectric Sphere

An experiment is set up to measure the electric potential and field within a uniformly polarized die

Electric Potential and Potential Energy in a System of Charges

This question focuses on calculating electric potentials and potential differences created by point

Electric Potential from a Point Charge

A point charge $$Q = +3.0*10^{-9} \; C$$ is located at the origin. Answer the following:

Electric Potential from a Uniformly Charged Rod

This problem requires you to derive the electric potential at a point on the perpendicular bisector

Electric Potential of a Point Charge: Experimental Investigation

A researcher is studying the electric potential generated by a point charge and comparing theoretica

Electrostatic Equilibrium in Concentric Spherical Shells

Two concentric spherical shells are assembled: an inner conducting sphere of radius R_inner carrying

Electrostatic Equilibrium in Conductors

A researcher investigates the behavior of electric charges in a conductor connected to a voltage sou

Electrostatic Force in an Equilateral Triangle of Charges

This problem examines the net electrostatic force on a charge located at one vertex of an equilatera

Evaluating the Effects of Dielectric Materials on a Capacitor

In this problem, you will derive the capacitance of a parallel plate capacitor both in vacuum and wh

Experimental Determination of Electric Field Gradient Near a Dipole

Devise an experiment to measure the spatial variation of the electric field near an electric dipole.

Experimental Verification of Coulomb's Law

A student conducts an experiment to verify Coulomb's law by measuring the electrostatic force betwee

Field and Potential from an Infinite Sheet of Charge

An infinite sheet has a uniform surface charge density of $$\sigma = 4.0 \times 10^{-6} \; C/m^2$$.

Field Inside and Outside a Charged Sphere

A non-conducting sphere of radius $$R = 0.20 \; m$$ carries a uniformly distributed charge of $$Q =

FRQ 4: Analysis of Non-Uniform Electric Field in a Laboratory Setup

A student reports that their electric field measurements in a region with multiple charge sources sh

FRQ 9: Evaluation of Battery Charging Data Using Electric Potential Analysis

A media report on a new battery technology claims that the voltage output decays linearly during the

FRQ 11: Conservation of Charge in Circuit Analysis

A media report on a simple electrical circuit claims that during transient states, charge conservati

FRQ 14: Analysis of Electrostatic Force Variation with Distance

A researcher claims that the measured electrostatic force between two point charges precisely follow

FRQ 19: Evaluation of Electric Field Line Distribution Claims

A recent report suggests that the density of electric field lines in simulation images accurately re

Gauss' Law and Cylindrical Charge Distributions

Consider an infinite line of charge with linear charge density $$\lambda$$ and a cylindrical shell w

Gauss' Law for a Non-Uniform Spherical Charge Distribution

Consider a sphere of radius R with a radial dependent charge density given by $$\rho(r) = \rho_0*(r/

Gauss' Law: Spherical Charge Distribution

A point charge is enclosed within a spherical surface. Analyze the electric flux through the sphere

Mapping Electric Field in a Complex Geometry via Monte Carlo Integration

An experiment is performed to map the electric field in a region with a complex charge distribution

Mapping the Electric Field of a Charged Sphere

Design an experiment to map the electric field around a uniformly charged insulating sphere using a

Multi-Charge Interaction Analysis

Three point charges are located at the vertices of an equilateral triangle with side length $$0.10 \

Non-Uniform Charge Distribution in a Sphere

Consider a sphere of radius $$R$$ in which the volume charge density varies with radius as $$\rho(r)

Optimization of Non-uniform Charge Distribution

A researcher aims to optimize the electric field at a point P on the y-axis (distance $$d$$ from the

Potential from a Line of Charge with Variable Density

A rod extends along the x-axis from $$x = 0$$ to $$x = 0.8 \; m$$. Its linear charge density is give

Potential on the Axis of a Charged Disk

A thin circular disk of radius $$R = 0.15 \; m$$ has a uniform surface charge density of $$\sigma =

Superposition of Electric Potentials

This problem involves calculating the net electric potential at a point due to two point charges usi

Surface Charge Distribution on an Irregular Conductor

A conductor has a non-uniform surface charge density given by $$\sigma(\theta) = \sigma_0 \sin(\thet

Uniformly Charged Sphere: Electric Field and Potential

A sphere of radius $$R$$ carries a uniform charge distribution with a total charge $$Q$$. Analyze th

Work Done by a Uniform Electric Field

This problem examines the work done on a charged particle as it moves in a uniform electric field an

Analysis of Capacitor Leakage Current and Its Impact on Stored Energy

Capacitor leakage current, modeled as a resistor in parallel with a capacitor, causes the stored ene

Analyzing Dielectric Constant Influence on Capacitance

When a dielectric is inserted into a capacitor, its capacitance increases. Consider the following ta

Calculus Analysis of Electric Field and Potential in a Non-Uniform Capacitor

A parallel plate capacitor contains a non-uniform dielectric where the permittivity varies with dist

Capacitance of Cylindrical Conductors

Two long, coaxial cylinders act as a capacitor. The inner cylinder has radius $$a$$ and the outer cy

Capacitor as a Pressure Sensor

A parallel plate capacitor is used as a pressure sensor. When pressure is applied, one of the plates

Capacitor Charging in an RC Circuit with Constant Resistance

A capacitor with capacitance $$C$$ is connected in series with a resistor of resistance $$R$$ and a

Capacitor Discharge Curve and Time Constant Computation

A capacitor in an RC circuit discharges over time, and its voltage follows the equation $$V(t)=V_0 e

Charge Distribution on a Complex Conductor

A complex-shaped conductor is studied to understand how its geometry affects surface charge density.

Comparative Analysis of Capacitor Types in High-Frequency Applications

Different types of capacitors (ceramic, electrolytic, and film) exhibit varying performance characte

Complex Impedance Analysis in a Capacitive Circuit

A series circuit consists of a resistor $$R$$, an inductor $$L$$, and a capacitor $$C$$ connected to

Design and Analysis of a Capacitor Bank for Energy Storage

For a high-energy application, a capacitor bank must be designed to store 1000 joules of energy at 1

Design and Analysis of a Tunable Capacitor Circuit for Frequency Modulation

A tunable capacitor is integrated into an LC oscillator circuit to modulate the produced frequency.

Design Optimization of a High-Energy Capacitor

Designers want to maximize the energy storage in a capacitor. The energy density is given by $$u = \

Designing a Faraday Cage for Sensitive Instrumentation

A sensitive piece of instrumentation must be shielded from external electromagnetic interference. Yo

Determining Dielectric Constant Experimentally

In an experiment, a set of identical parallel plate capacitors is assembled with different dielectri

Determining Permittivity from Parallel Plate Capacitor Data

An experiment involving a parallel plate capacitor was conducted to determine the permittivity of th

Dielectric Breakdown Analysis

Dielectric breakdown occurs when a dielectric material is exposed to an electric field that exceeds

Dielectric Comparison in Capacitors

Two parallel plate capacitors of identical geometry are constructed. Capacitor A uses glass (with a

Electric Field Distribution on a Conductor

Consider an isolated conductor with a non‐uniform surface curvature. The local charge density, and h

Electric Shielding Efficiency in Faraday Cages

A researcher claims that a Faraday cage constructed with aluminum panels significantly reduces exter

Electrostatic Pressure on a Conductor

When a conductor is charged, an electrostatic pressure is developed on its surface. Answer the follo

Energy Balance in a Non-Ideal RC Circuit

Consider an RC circuit in which a capacitor discharges through a non-ideal resistor whose resistance

Energy Extraction from Capacitor Discharge in a Flash Circuit

A camera flash circuit utilizes a capacitor that rapidly discharges to produce a flash. Consider a c

Energy Stored in a Parallel Plate Capacitor - Calculus Based Derivation

A parallel plate capacitor with plate area \(A\) and plate separation \(d\) is filled with a dielect

Evaluation of Faraday Cage Performance under High-Frequency EMI

A Faraday cage’s performance is tested against high-frequency electromagnetic interference. Using th

Experimental Analysis of a Faraday Cage

An experiment is conducted to evaluate the shielding effectiveness of a Faraday cage. A graph of the

Experimental Determination of Dielectric Constant Using a Variable Capacitor

Design an experiment to determine the dielectric constant \(k\) of an unknown material using a varia

Faraday Cage Effect and Field Analysis

A Faraday cage is constructed using a metal mesh configuration. Analyze the underlying principles th

FRQ 3: Charging and Discharging of a Parallel Plate Capacitor

In a laboratory experiment, students measure the voltage across a parallel plate capacitor in an RC

FRQ 5: Capacitor Energy Storage Measurement with Voltage Ramp Experiment

A student charged a capacitor using a voltage source that increased linearly with time, intending to

FRQ 5: Faraday Cage Performance Analysis

A Faraday cage is constructed to protect sensitive electronic equipment from electromagnetic pulses

FRQ 6: Measurement of Surface Charge Density on a Conductor

An experiment was designed to map the surface charge density on an irregularly shaped conductor. The

FRQ 8: Dielectric Breakdown Analysis under Variable Frequencies

A dielectric material is subjected to an applied voltage at various frequencies. Experimental data s

FRQ 12: Energy Storage in Capacitors Under Varying Voltages

An experiment is performed in which the energy stored in a capacitor is measured at different applie

FRQ 13: Theoretical Analysis of Electric Field Shielding in Layered Conductors

Multiple concentric conductive shells are used to shield a sensitive instrument from external electr

FRQ 14: Curvature Effects in Conductors: Spherical vs. Elongated Shapes

An experiment compares the electric field intensity at the surface of a spherical conductor with tha

FRQ 16: Measurement of Electric Shielding Effectiveness Using a Conductive Film

An experiment was set up to measure the effectiveness of an electric shield composed of a thin condu

FRQ 18: Apparent Charge Redistribution in a Connected Conductor System

A researcher investigated charge redistribution between two conductors of different geometries that

FRQ 19: Temperature Effects on Capacitor Performance

A controlled experiment investigates how the capacitance of a capacitor varies with temperature. The

Impact of Dielectric Constant on Capacitance

The capacitance of a capacitor is influenced by the dielectric constant of the material between its

Influence of Plate Separation on Capacitance

The capacitance of a parallel plate capacitor is inversely proportional to the distance between its

Investigating the Role of Curvature on Surface Charge Density in Conductors

A researcher studies a set of conductors with different curvatures (from flat plates to spheres) and

Investigation of a Variable Capacitor

A variable capacitor is tested by adjusting its knob position, and the corresponding capacitance is

Measurement of Electric Field on a Conductor's Surface

A researcher wants to measure the electric field just outside the surface of a spherical conductor.

Modeling a Variable Capacitor in a Tuning Circuit

A variable capacitor in a radio tuning circuit adjusts its capacitance by rotating one of its plates

Parallel Plate Capacitor with Gradually Varying Dielectric

Consider a parallel plate capacitor with plate area $$A$$ and separation $$d$$, where the dielectric

RC Discharge Experimental Data Analysis

Analyzing the discharge of a capacitor through a resistor is critical for understanding RC circuits.

Surface Charge Density and Electric Field on a Conductor

Consider a conductor in electrostatic equilibrium. Use Gauss' law to relate the surface charge densi

Surface Charge Density on a Non-Uniform Conductor

A conductor with regions of differing curvature will have a non-uniform surface charge density $$\si

Temperature Effects on Capacitor Dielectric Properties

The dielectric constant of a capacitor’s insulating material can vary with temperature. Suppose the

Time-Dependent Dielectric Response in a Capacitor

In a capacitor, the inserted dielectric experiences a change in its properties over time due to temp

Time-Varying Voltage Capacitor System

A capacitor of capacitance $$C$$ (with a dielectric) is connected to a voltage source that varies wi

Transient Response in an RC Circuit with Variable Resistance

Consider an RC circuit in which the resistor’s value varies with time as $$R(t)=R_0\,(1+\alpha t)$$,

Variable Dielectric in a Capacitor

A capacitor is made with a dielectric whose property varies along the separation between the plates.

AC Circuit: Impedance and Power Factor Analysis

An AC circuit operates at a frequency of $$f = 60$$ Hz and consists of a resistor (R = $$10$$ $$\Ome

Analysis of a Non-Ideal Battery's Internal Resistance

A battery’s terminal voltage was measured under various load currents to study the effect of interna

Analysis of a Triangular Wave from a Function Generator

A function generator outputs a triangular wave defined by $$V(t)=12*(2*abs((t \mod 1)-0.5)-1)$$, whe

Analysis of Non-Linear Temperature-Dependent Resistors

Some resistors exhibit a non-linear, exponential dependence on temperature. Analyze such a resistor

Analysis of Power Supply Stability

Examine a power supply circuit where the output voltage is affected by varying load conditions. Use

Analysis of Resonance in an RLC Circuit

Consider a series RLC circuit with R = 10 Ω, L = 0.5 H, and C = 20 μF. Analyze the circuit’s behavio

Analyzing Complex Circuits using Kirchhoff’s Laws

This question focuses on applying Kirchhoff’s Voltage and Current Laws to analyze a complex circuit

Analyzing the Effect of Temperature on Resistivity in a Circuit

A conductor's resistance varies with temperature according to the formula $$R = R_0 (1+ \alpha (T-T_

Capacitance in Complex Networks

In a circuit containing both series and parallel capacitors, a claim is made that the total capacita

Circuit Design Challenge: Maximizing Power Transfer

A circuit is designed to maximize power transfer to a load resistor. Using the Thevenin equivalent c

Design of an AC to DC Power Supply Circuit

An AC power supply circuit is designed using a transformer, a rectifier, and a filter capacitor to p

Discharging Capacitor Experiment with Ignored Resistance Variation

A researcher investigates the discharging behavior of a capacitor in an RC circuit by disconnecting

Electrolytic Capacitor Behavior Experiment with Misinterpretation of Dielectric Absorption

An experiment is carried out to measure the charging and discharging curves of an electrolytic capac

Electromotive Force and Internal Resistance Investigations

A battery with an electromotive force (EMF) of 9 V and an internal resistance of 1 \(\Omega\) is con

Evaluating Oscilloscope Measurement Accuracy in AC Circuits

During an experiment, voltage waveforms captured by an oscilloscope showed slight distortions compar

Frequency Dependence of Capacitive Reactance

Investigate how the capacitive reactance in an AC circuit depends on frequency, and use calculus to

FRQ 2: Series Circuit Analysis

A circuit consists of four resistors connected in series with values $$R_1 = 2\,\Omega$$, $$R_2 = 3\

FRQ 7: Kirchhoff's Voltage Law in a Multi-Loop Circuit

A circuit consists of two loops that share a common resistor. The circuit has two voltage sources: $

FRQ 8: Kirchhoff's Current Law at a Node

At a junction in a circuit, three branches converge. Two of the branches have measured currents of $

FRQ 16: Circuit Measurement Techniques

In a laboratory setting, students are asked to measure voltage, current, and resistance in a circuit

FRQ 19: Experimental Design: Measuring Capacitance Using an RC Circuit

Design an experiment to measure the capacitance of an unknown capacitor using its charging curve in

Function Generator Misconfiguration in AC Circuit Analysis

An experiment is conducted to study the response of an RLC circuit to an AC signal generated by a fu

Impact of Temperature on Resistance

Resistance in a conductor can change with temperature. Investigate this variation and its implicatio

Investigating Temperature-Dependent Resistance of a Thermistor

This question explores how the resistance of a thermistor varies with temperature, requiring both ma

Kirchhoff's Voltage Law in a Complex Circuit

Consider a circuit where three resistors (R1, R2, and R3) are connected in series with a battery of

Kirchhoff's Voltage Law in a Multi-Loop Circuit

A multi-loop circuit contains a 15 V battery and three resistors with values R1 = 3 Ω, R2 = 4 Ω, and

Low-Pass Filter Analysis in an RC Circuit

An RC circuit is configured as a low-pass filter. Analyze its transfer function and cutoff frequency

Measurement of Resistor's Temperature Coefficient with Overlooked Self-Heating

A researcher sets up an experiment to determine the temperature coefficient of a resistor by applyin

Non-Ideal Battery and Internal Resistance Effects

A researcher examines a battery having an open-circuit electromotive force (EMF) of $$12\,V$$. Howev

Non-Ideal Battery Effects on Circuit Performance

A researcher investigates a circuit in which a battery with EMF E and internal resistance r is conne

Oscilloscope Analysis of an AC Signal

An oscilloscope displays the waveform of an AC signal. Use the provided trace to analyze its charact

Parallel Circuit Analysis with Variable Resistor

Consider a circuit with three resistors connected in parallel to an 18 V battery. The resistors have

Power Calculation in Circuit Experiment with Misinterpretation of Resistive vs Reactive Power

A student performs an experiment to determine the power consumed in an AC circuit containing resisti

Power Dissipation in a Resistor

Consider a resistor of $$R = 10\,\Omega$$ connected to a time-varying voltage source given by $$V(t)

RC Circuit Charging Analysis

Consider an RC circuit where a resistor $$R = 10\,k\Omega$$ and a capacitor $$C = 100\,\mu F$$ are c

RC Discharging Circuit Analysis

A capacitor initially charged to $$20\,V$$ discharges through a resistor with $$R=5000\,\Omega$$ and

RC Oscillator Frequency Determination with Ignored Parasitic Capacitance

A student designs an RC oscillator intended to generate a periodic signal at a target frequency. The

Sensitivity Analysis of a Series-Parallel Circuit

This question explores how variations in a component affect the overall resistance of a circuit that

Series Circuit Analysis and Ohm's Law

A student conducted an experiment on a series circuit powered by a constant 12 V supply with differe

Stability Analysis of an RC Oscillator

An RC oscillator generates oscillations with a frequency given by $$f = \frac{1}{2\pi R C}$$. Due to

Temperature Effect on Thermistor Resistance

A thermistor in a circuit has a resistance given by $$R(T)=1000*\exp(0.01*(T-25))$$, where T is the

Temperature Effects on Resistance in a Circuit

In many practical circuits, resistor values vary with temperature. Analyze the effect of temperature

Temperature Effects on Resistance in a Voltage Divider

A thermistor with a resistance that varies with temperature according to $$R(T)=R_0(1+\alpha T)$$, w

Transient Analysis in an RL Circuit

A circuit comprising a resistor and an inductor (an RL circuit) is subject to a DC voltage step. Ana

Transient Response in an RL Circuit with Neglected Inductive Kickback Effects

A student investigates the transient response of an RL circuit by abruptly opening a switch and obse

Transient Response in RC Charging Circuit

In an RC circuit with resistor R = 10 Ω and capacitor C = 100 μF, the capacitor is initially uncharg

Variable Resistor and Thermistor Behavior Analysis

A variable resistor in a circuit is adjusted to change its voltage drop linearly with rotation. Addi

Analyzing a Magnetic Resonance Experiment

In a magnetic resonance experiment, a sample is placed in a uniform static magnetic field of 1.5 T a

Assessment of Magnetic Field Variations in Coaxial Circular Loops

A system consists of multiple coaxial circular loops of wire, each with different radii, carrying id

Calculus in Magnetic Induction: Time-Varying Current Effects

A solenoid with 400 turns over a length of 0.8 m is driven by a time-varying current expressed as $$

Calculus of a Finite Solenoid’s Magnetic Field Variation

A solenoid of finite length exhibits a non-uniform turn distribution along its length. (a) Derive

Calculus-Based Derivation of Solenoid Magnetic Field using Ampère's Law

A theoretical paper claims to derive the uniform magnetic field inside an ideal solenoid using Ampèr

Comparison of Magnetic Fields Inside and Outside a Solenoid

Measurements were taken of the magnetic field in and around a long solenoid. The data in the table b

Critical Analysis of Ampère's Law in a Non-Uniform Magnetic Field

An experiment uses a loop to measure the magnetic field in a region where the field is non-uniform.

Critical Evaluation of Magnetic Field Uniformity in Solenoid Experiments

A recent experimental study claims that the magnetic field inside a solenoid is perfectly uniform, a

Determining Torsional Stiffness from Magnetic Torque Data

A thin wire is subjected to magnetic torque in a uniform magnetic field, resulting in an angular dis

Effect of Loop Area on Solenoid Magnetic Field

In an experimental study, a researcher examines how the area of a loop placed inside a solenoid affe

Experimental Investigation of Force Between Parallel Current-Carrying Wires

An experiment is conducted to verify the force between two parallel current-carrying wires. The wire

Experimental Validation of the Right-Hand Rule

An experiment is designed to validate the right-hand rule by measuring the direction of the magnetic

Force Between Parallel Current-Carrying Wires

Two long, parallel wires separated by a distance \(d\) carry currents \(I_1\) and \(I_2\). They exer

Force Between Two Parallel Current-Carrying Wires

Two parallel wires carry steady currents and interact magnetically. Analyze their interaction using

Force on a Current-Carrying Wire

Investigate the force experienced by a current-carrying wire in a uniform magnetic field and analyze

Force on a Current-Carrying Wire at an Angle in a Magnetic Field

A 0.30 m long straight wire carrying a current of 5.0 A is placed in a uniform magnetic field of 0.2

Integral Calculation of the Magnetic Field from a Finite Straight Wire

Consider a finite straight wire of length $$L$$ carrying a steady current $$I$$. Using calculus, der

Investigating the Magnetic Field Gradient in a Toroid

A toroid is constructed with a mean radius of 0.20 m, 300 turns, and carries a current of 2 A. (a) D

Magnetic Field Decay with Distance in a Solenoid

An experiment measures the magnetic field both inside and outside a solenoid as a function of distan

Magnetic Field Distribution around a Bar Magnet

A bar magnet produces a magnetic field that is strongest near its poles. Explore the variation in fi

Magnetic Field in a Solenoid

Consider a solenoid of length L with N turns carrying a current I. (a) Derive the expression for th

Magnetic Field in a Toroid

A toroid is constructed by winding a wire into a doughnut-shaped coil with N turns. The toroid has a

Magnetic Field Inside a Solenoid

An experiment is designed to measure the magnetic field inside a solenoid. The solenoid’s field is h

Magnetic Field Inside a Solenoid

A solenoid is a coil of wire with a large number of turns. Consider a solenoid of length $$L$$, with

Magnetic Field Inside a Solenoid using Ampère's Law

A long solenoid with a uniform turn density (n turns per unit length) carries a current $$I$$. Using

Magnetic Field Mapping of a Bar Magnet

A researcher collects experimental data measuring the magnetic field strength at various distances f

Magnetic Field Measurement around a Straight Conductor

An experimental setup is constructed to measure the magnetic field around a long straight conductor

Magnetic Field Measurement from a Straight Conductor

A researcher seeks to verify the theoretical dependence of the magnetic field on the distance from a

Magnetic Field of a Circular Loop

A single circular loop of wire with radius $$R$$ carries a steady current $$I$$. The magnetic field

Magnetic Field of a Straight Conductor

Consider an infinitely long straight wire carrying a constant current. Use the standard formula deri

Magnetic Field Strength Inside a Solenoid with Varying Coil Turns

A research report states that the magnetic field inside a solenoid increases linearly with the numbe

Magnetic Force in a Particle Detector

In a particle detector, a charged particle with mass $$m$$, charge $$q$$, and velocity $$v$$ moves p

Magnetic Force on a Moving Charged Particle

In a controlled laboratory experiment, a charged particle with charge $$q = 2\times10^{-6} \; C$$ an

Magnetic Potential in a Current-Free Region

In regions without free currents, the magnetic potential $$\phi$$ satisfies Laplace’s equation. Deri

Net Force on a Circular Current Loop in a Non-Uniform Magnetic Field

A circular current loop of radius $$R$$ is placed in a non-uniform magnetic field given by $$B(r)=\f

Optimization of MRI Magnetic Field Strength

An MRI machine design requires a highly uniform and strong magnetic field, generated by a solenoid.

Optimization of Solenoid Design

An engineer is designing a solenoid for an MRI machine with a fixed length of wire. Use calculus to

Spatial Mapping of Magnetic Field Vectors in a Wire Array

An array of parallel current-carrying wires is arranged, and a 3D Hall probe is used to measure the

Torque on a Current-Carrying Loop

In the development of electric motors, the phenomenon of torque acting on current-carrying loops is

Velocity Selection in a Particle Detector

A particle detector is designed using crossed uniform electric $$E$$ and magnetic $$B$$ fields that

Verification of Ampère’s Law through Closed-Loop Measurements

In an experiment, the line integral of the magnetic field around a closed circular loop was measured

AC Generator and Induced Current Analysis

An AC generator produces a time-varying EMF described by $$\epsilon(t)= 50*\cos(100*t)$$ (volts). Co

Analysis of Ampere's Law in a Capacitor Charging Circuit

An experiment is performed where a loop of wire is placed around a charging capacitor to measure the

Analysis of Lenz's Law via a Coil and Falling Magnet Experiment

In an experiment, a magnet is dropped through a copper tube and coils wrapped around the tube record

Analyzing Back EMF in an LR Circuit

Consider an LR circuit where, after a steady state current has been established, the applied voltage

Application of Ampere's Law with Maxwell Correction in a Capacitor System

This problem applies Ampere's Law with Maxwell's correction to analyze the displacement current in a

Application of Gauss's Law to a Charged Sphere

A solid sphere with total charge $$Q$$ is uniformly charged throughout its volume.

Application of Maxwell's Equations in a Capacitor Circuit

A parallel-plate capacitor is charged by a time-varying current. Experimental measurements of the el

Application of Maxwell's Equations in Dynamic Field Situations

Consider a region in which both electric and magnetic fields vary with time. Maxwell's equations gov

Calculating Coiled Inductance via Magnetic Flux

A solenoid with $$N$$ turns has a cross-sectional area $$A$$ and is placed in a uniform magnetic fie

Changing Current and Induced Voltage in a Coil

A solenoidal coil with $$N$$ turns and cross-sectional area $$A$$ is energized such that the current

Comprehensive Analysis of an RLC Circuit

An RLC circuit consisting of a resistor $$R$$, inductor $$L$$, and capacitor $$C$$ is subjected to a

Damped Oscillations in a Practical LC Circuit

A real LC circuit contains a small resistance R, causing damping of its oscillations. Consider an LC

Deriving Transformer Efficiency via Energy Calculations

A transformer transfers energy from its primary coil to its secondary coil. Assume that the input an

Design of a Magnetic Levitation and Induction Heating System for Transportation

A research team is developing a hybrid transportation system that utilizes electromagnetic induction

Displacement Current in a Charging Capacitor

A researcher investigates the role of displacement current in a capacitor circuit being charged by a

Eddy Current Braking Analysis

A rotating metallic disk is subjected to a magnetic field, and eddy currents are induced which produ

Effect of Coil Turns on Induced EMF

A researcher varies the number of turns in a coil while keeping the rate of change of magnetic flux

Electromagnetic Generator Analysis

A simple AC generator consists of a rectangular loop with area $$A=0.10\;m^2$$ rotating in a uniform

Electromagnetic Induction in a Moving Conductor

A straight conductor of length $$l$$ moves with a constant velocity $$v$$ perpendicular to a uniform

Electromagnetic Induction Sensor Design

Design a magnetic sensor based on electromagnetic induction that can detect small variations in a ma

Electromagnetic Oscillator Simulation

A circuit is designed incorporating an LC oscillator with a resistor to simulate damping, resulting

Energy Transfer in LC Oscillations

An LC circuit oscillates with an initial charge $$Q_0$$ on the capacitor. The circuit consists of an

Faraday's Law and a Sliding Bar on Rails

A metal bar of length $$l$$ slides on a pair of parallel, conducting rails with a constant velocity

Induction Heating and Eddy Currents

In an induction heating device, a time-varying magnetic field is produced by an alternating current

Inductive Kickback and Energy Dissipation in an R-L Circuit

A researcher studies the phenomenon of inductive kickback in an R-L circuit when the current is sudd

Inductor Energy Storage Calculation

Consider an inductor with an inductance of $$L = 0.5$$ H carrying a current that increases according

Interpreting a Visual Diagram of Magnetic Dipole Fields

A diagram of a bar magnet with magnetic field lines is provided. Analyze the diagram based on the fo

Investigating Eddy Currents in a Conducting Tube

A magnet is dropped through a non-magnetic, conducting tube. The induced eddy currents in the tube g

Investigating Induced EMF in a Rotating Loop

A conducting loop of area $$A$$ rotates in a uniform magnetic field $$B$$ with constant angular velo

Investigation of EMF in a Moving Conductor

An experiment was designed to investigate the electromotive force (EMF) induced in a straight conduc

LR Circuit Transient Response

Consider an LR circuit comprising an inductor with inductance $$L$$ and a resistor of resistance $$R

Magnetic Field Calculation Inside a Solenoid

This problem requires the use of Ampere's Law to determine the magnetic field inside a long solenoid

Magnetic Field Distribution and Gauss's Law

A uniformly charged sphere produces an electric field that varies with the distance from the center.

Magnetic Field Mapping using Gauss’s Law

In a laboratory setup, a spherical closed surface is used to measure the net magnetic flux produced

Magnetic Field Measurement in a Solenoid

An experiment is carried out to map the magnetic field inside a long solenoid using a Hall probe. Th

Magnetic Flux Leakage in Practical Circuits

Magnetic flux leakage is measured on a circuit board using sensors placed along its length. The tabl

Magnetic Flux Variation Over Time

A coil is subjected to a time-varying magnetic field. The magnetic flux $$\Phi(t)$$ through the coil

Magnetic Levitation Analysis

Evaluate a magnetic levitation system where an object of mass $$m$$ is levitated by electromagnetic

Magnetic Levitation and Electromagnetic Braking

Magnetic levitation (maglev) trains use electromagnetic induction both to levitate above the tracks

Magnetic Levitation and Induced Currents

An object with mass $$m$$ is levitated above a conducting plate due to induced eddy currents generat

Magnetic Levitation Using Electromagnetic Induction

A magnetic levitation (maglev) system uses electromagnets to levitate a train. As the train moves, a

Maxwell’s Equations – Gauss’s Law

Gauss’s Law is one of Maxwell’s equations and relates the electric flux through a closed surface to

Measuring Magnetic Flux Variations

A researcher rotates a coil in a uniform magnetic field to study how the magnetic flux changes with

Resonant Frequency in an LC Circuit

An LC circuit is constructed with an inductor $$L=0.1\;H$$ and a capacitor $$C=10\;\mu F$$.

Resonant Frequency in LC Circuits

This problem involves the analysis of an LC circuit and the determination of its resonant frequency.

Rotating Loop Generator Analysis

A researcher designs an experiment using a rotating rectangular loop in a uniform magnetic field to

Sinusoidal EMF from an Alternating Current Generator

A researcher investigates an AC generator where a coil rotates in a magnetic field, producing a sinu

Sliding Conductor in a Magnetic Field

A researcher investigates electromagnetic induction by moving a straight conductor of length $$l$$ a

Time-Dependent Magnetic Flux in a Rotating Coil

A coil of area $$A$$ rotates in a constant magnetic field $$B$$ with a constant angular velocity $$\

Time-Varying Magnetic Fields and Induced Currents

A solenoid with $$N$$ turns and cross-sectional area $$A$$ is subjected to a magnetic field that dec

Time-Varying Magnetic Fields and Induced Currents in a Solenoid

A long solenoid is subjected to an external magnetic field varying with time as $$B(t) = B_0 \cos(\o

Transformer Design and Analysis

A transformer is designed to step down an input voltage $$V_{p}$$ to an output voltage $$V_{s}$$. Th

Transient Analysis in an LR Circuit

An LR circuit consists of an inductor with $$L=2\;H$$ and a resistor with $$R=4\;\Omega$$ connected

Transient Behavior Analysis in an LR Circuit

A researcher examines the transient current behavior in an LR circuit immediately after the switch i

Transient LR Circuit Response

A series LR circuit is connected to a battery at $$t = 0$$, and the current begins to rise exponenti