Source: UIUC Electricity, Magnetism, and Thermodynamics Final Examination
Tags: magnetic force, Lorentz force, solenoid, magnetic field, self-inductance, Faraday's law, Lenz's law, induced EMF, RL circuit, RC circuit, time constant, LC oscillations, damped RLC circuit, Q-factor
Difficulty: Intermediate | Prerequisites: Electrostatics and capacitors notes (Part 1), cross product, basic differential equations (or at least comfort with exponential growth/decay).
Once you move from static charges to moving charges, magnetism enters the picture. A current-carrying wire produces a magnetic field, a magnetic field exerts a force on moving charges, and a changing magnetic field induces an EMF. These ideas, wrapped up in Faraday's law and Lenz's law, are the basis for electric generators, transformers, and every form of electromagnetic technology. The circuit elements that emerge from this, inductors alongside the resistors and capacitors you already know, produce the time-dependent behaviour (exponential charging, oscillations, damping) that dominates the second half of a typical E&M course.
The magnetic force on a moving charge depends on the cross product of velocity and field, so it vanishes when the two are parallel. Solenoids create uniform internal magnetic fields proportional to current and turns per unit length. Inductors resist changes in current, giving rise to exponential transients in RL and RC circuits and to oscillations in LC circuits. Resistance in an RLC circuit causes those oscillations to decay.
Magnetic Force (Lorentz Force)
The force on a charge q moving with velocity v through a magnetic field B is F = qv × B. Its magnitude is F = qvB sin θ, where θ is the angle between v and B. In simple terms, the force is largest when the charge moves perpendicular to the field and zero when it moves parallel to it.
Solenoid
A long, tightly wound coil of wire. Inside an ideal (infinitely long) solenoid, the magnetic field is uniform: B = μ₀nI, where n is the number of turns per unit length and I is the current. Think of it as a way to create a nearly uniform magnetic field in a controlled region.
Self-Inductance (L)
A measure of how much magnetic flux a coil generates through itself per unit current: L = NΦ_B / I. Measured in henrys (H). In simple terms, inductance tells you how strongly a coil resists changes in the current flowing through it.
Faraday's Law of Induction
The induced EMF in a loop equals the negative rate of change of magnetic flux through that loop: ε = −dΦ_B/dt. Think of it as nature's way of generating voltage whenever a magnetic environment changes.
Lenz's Law
The direction of the induced current is always such that it opposes the change in magnetic flux that produced it. In simple terms, the induced current creates its own magnetic field that tries to keep things the way they were.
Time Constant (τ)
For an RC circuit, τ = RC. For an RL circuit, τ = L/R. After one time constant, the charging quantity reaches about 63% of its final value (or a decaying quantity drops to about 37%). Think of it as the characteristic "speed" of the circuit's response.
LC Angular Frequency (ω₀)
The natural oscillation frequency of an LC circuit: ω₀ = 1/√(LC). In simple terms, this is the rate at which energy bounces back and forth between the capacitor and the inductor.
Q-Factor (Quality Factor)
A dimensionless number that describes how underdamped an RLC oscillator is. A high Q means sharp resonance and slow energy loss; a low Q means broad resonance and fast energy loss. Think of it as a measure of how "ringy" the circuit is: a bell has a high Q, a thud has a low Q.
The force F = qv × B is always perpendicular to both v and B.
Because the force is perpendicular to the velocity, it does no work on the particle. A magnetic field cannot change a particle's kinetic energy; it can only change the direction of motion.
When v is parallel to B (θ = 0° or 180°), sin θ = 0, so the magnetic force is zero. The particle travels in a straight line along the field.
When v is perpendicular to B, the force provides centripetal acceleration, and the particle moves in a circle of radius R = mv / (qB).
A proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C) is accelerated from rest through V = 500 V, then enters a region where B = 0.1 T perpendicular to its velocity.
Kinetic energy gained: KE = qV = (1.6 × 10⁻¹⁹)(500) = 8.0 × 10⁻¹⁷ J.
Speed: v = √(2KE/m) = √(2 × 8.0 × 10⁻¹⁷ / 1.67 × 10⁻²⁷) ≈ 3.09 × 10⁵ m/s.
Radius: R = mv/(qB) = (1.67 × 10⁻²⁷)(3.09 × 10⁵) / ((1.6 × 10⁻¹⁹)(0.1)) ≈ 3.2 × 10⁻² m ≈ 3.2 cm.
The path is a circle because the magnetic force is always perpendicular to the velocity.
B = μ₀nI inside an ideal solenoid.
If the current is doubled and the turns per unit length are halved, B = μ₀(n/2)(2I) = μ₀nI. The field remains unchanged.
The field outside an ideal solenoid is approximately zero.
L = μ₀n²Al, where A is the cross-sectional area and l is the length.
Inserting a ferromagnetic core replaces μ₀ with μ = κ_m μ₀ (where κ_m >> 1), dramatically increasing L.
Increasing the cross-sectional area, the number of turns, or the core permeability all increase L. Resistance of the wire does not appear in the inductance formula.
When the switch closes at t = 0, the inductor initially opposes any change in current, so I(0) = 0.
The current grows exponentially toward its maximum value I_max = V/R:
I(t) = (V/R)(1 − e^(−Rt/L))
The time constant is τ = L/R. At t = τ, the current has reached about 63% of I_max.
To find the time to reach 50% of I_max: 0.5 = 1 − e^(−t/τ), so e^(−t/τ) = 0.5, giving t = τ ln 2 ≈ 0.693τ.
V = 12 V, R = 6 Ω, L = 0.3 H.
I_max = 12/6 = 2 A.
τ = L/R = 0.3/6 = 0.05 s.
I(t) = 2(1 − e^(−t/0.05)) A.
Time to reach 50% of I_max: t = 0.05 × ln 2 ≈ 0.0347 s ≈ 34.7 ms.
When charging through a resistor R from a battery V, the charge on the capacitor grows as Q(t) = CV(1 − e^(−t/RC)).
The time constant is τ = RC. At t = τ, the charge reaches about 63% of its maximum value Q_max = CV.
During discharge, Q(t) = Q₀ e^(−t/RC).
In an ideal (resistanceless) LC circuit, energy oscillates between the capacitor's electric field and the inductor's magnetic field.
The angular frequency is ω₀ = 1/√(LC).
For L = 10 mH and C = 25 μF: ω₀ = 1/√(0.01 × 25 × 10⁻⁶) = 1/√(2.5 × 10⁻⁷) = 1/(5 × 10⁻⁴) = 2000 rad/s.
The charge on the capacitor varies as Q(t) = Q_max cos(ω₀t + φ).
Adding resistance R to an LC circuit causes the oscillations to decay over time.
The rate of energy decay is determined by the resistance R. Higher R means faster decay.
The Q-factor is inversely related to damping: Q = ω₀L / R (for a series RLC).
A high Q-factor gives a narrow, sharp resonance peak.
A low Q-factor gives a wide, broad resonance peak.
The statement "a high Q-factor corresponds to a wide resonance peak" is false.
The induced current always flows in a direction that creates a magnetic field opposing the change in flux.
When a loop moves into a region of constant magnetic field, the flux through the loop increases, so the induced current creates a field that opposes that increase (opposing the change, not necessarily opposing the external field itself).
Lenz's law is a consequence of conservation of energy.
Quantity | Formula |
|---|---|
Magnetic force | F = qvB sin θ |
Cyclotron radius | R = mv / (qB) |
Solenoid field | B = μ₀nI |
Permeability of free space | μ₀ = 4π × 10⁻⁷ T·m/A |
Self-inductance (solenoid) | L = μ₀n²Al |
Faraday's law | ε = −dΦ_B / dt |
RL time constant | τ = L/R |
RL charging current | I(t) = (V/R)(1 − e^(−Rt/L)) |
RC time constant | τ = RC |
RC charging | Q(t) = CV(1 − e^(−t/RC)) |
LC angular frequency | ω₀ = 1/√(LC) |
Self-induced EMF | ε = −L (dI/dt) |
RL circuits are the basis of relay timing, spark suppression in automotive ignition systems, and the "kick" you feel from an inductive load when you switch it off. LC oscillations are the tuning mechanism in radio receivers: adjusting the capacitance shifts the resonant frequency to match different stations. MRI machines use enormous superconducting solenoids to produce the uniform magnetic fields needed for medical imaging. Lenz's law explains electromagnetic braking in trains and roller coasters, where a conducting fin moving through a magnetic field experiences a retarding force without any physical contact.
Students often think a magnetic field can do work on a charged particle. It cannot. The magnetic force is always perpendicular to the velocity, so it changes direction but not speed.
Students sometimes believe the induced current always opposes the external field. Lenz's law says the induced current opposes the change in flux, which is a different thing. If the flux is decreasing, the induced current supports the external field to try to maintain the flux.
A common error is thinking the current in an RL circuit jumps immediately to V/R when the switch closes. The inductor prevents instantaneous changes in current, so I(0) = 0 and the current rises exponentially.
Self-induced EMF opposes the change in current, not supports it. The statement "self-induced EMF supports the increasing current" is false.
⚠️ When velocity is parallel to the magnetic field, the force is zero. This is a classic exam question designed to test whether you remember the cross product.
⚠️ Know what happens at t = 0 in an RL circuit (I = 0) and at t = ∞ (I = V/R). The exam tests both limits.
⚠️ The time constant τ = RC for an RC circuit and τ = L/R for an RL circuit. At t = τ, the charging quantity reaches 63% of its final value. This 63% figure appears repeatedly.
⚠️ In a damped RLC circuit, the resistance R determines the damping rate. The exam specifically asks which quantity controls energy decay, and the answer is R.
⚠️ A high Q-factor means a narrow resonance peak, not a wide one. This is a common true/false trap.
Fill in the blank: The magnetic force on a charged particle moving parallel to a magnetic field is ______.
True or false: An inductor allows current to change instantaneously.
Fill in the blank: In an RL circuit, the current reaches 63% of its maximum value after a time equal to ______.
True or false: Self-induced EMF in an inductor supports the direction of increasing current.
Fill in the blank: The angular frequency of an LC circuit with L = 10 mH and C = 25 μF is ______ rad/s.
Answers: 1. Zero. 2. False. 3. τ = L/R (one time constant). 4. False (it opposes the change). 5. 2000 rad/s.
Q: A particle with charge q moves with velocity v parallel to a uniform magnetic field B. What is the magnitude of the magnetic force?
A: Zero. F = qvB sin θ, and when v is parallel to B, θ = 0, so sin θ = 0.
Q: An infinitely long solenoid has n turns per unit length and carries current I. If the current is doubled and n is halved, what happens to B?
A: B = μ₀nI. The new field is μ₀(n/2)(2I) = μ₀nI. The field remains unchanged.
Q: Which of the following increases the self-inductance of a solenoid: decreasing area, increasing wire resistance, inserting a ferromagnetic core, or decreasing turns?
A: Inserting a ferromagnetic core. It replaces μ₀ with a much larger μ, increasing L.
Q: In an RL circuit with V = 12 V, R = 6 Ω, L = 0.3 H, what is the current immediately after the switch closes?
A: I(0) = 0. The inductor opposes any instantaneous change in current.
Q: Derive I(t) for the RL circuit above and find the time to reach 50% of I_max.
A: I(t) = (V/R)(1 − e^(−Rt/L)) = 2(1 − e^(−t/0.05)) A. For 50%: t = (L/R) ln 2 = 0.05 × 0.693 ≈ 0.035 s.
Q: In a damped RLC circuit, what physical quantity determines the rate of energy decay?
A: The resistance R.
Q: A loop of wire moves into a region of constant magnetic field. According to Lenz's law, what does the induced current do?
A: It creates a magnetic field that opposes the change in flux through the loop (opposing the increasing flux, not the external field itself).
Q: A proton accelerated through 500 V enters a 0.1 T field perpendicular to its velocity. What is the radius of its circular path?
A: v = √(2qV/m) ≈ 3.09 × 10⁵ m/s. R = mv/(qB) ≈ 3.2 cm.
The magnetic force on moving charges connects directly to the Hall effect, which is used to identify charge carriers in materials. Faraday's law is one of Maxwell's four equations and, together with the displacement current, leads to the prediction of electromagnetic waves (covered in Part 3). The exponential behaviour of RL and RC circuits reappears in the study of signal processing, filters, and the transient response of any linear system. LC oscillations are the electrical analogue of a mass on a spring, and adding resistance is analogous to adding friction.
Lorentz force, magnetic force on moving charge, cross product, right-hand rule, cyclotron motion, cyclotron radius, solenoid magnetic field, Ampere's law, self-inductance formula, ferromagnetic core, Faraday's law of induction, Lenz's law direction, electromagnetic induction, RL circuit transient, RC circuit charging, RC circuit discharging, time constant, exponential decay, LC circuit oscillation, angular frequency, natural frequency, damped oscillation, RLC circuit, quality factor, Q-factor, resonance peak width, energy decay rate