Difficulty: Intermediate | Prerequisites: Electrostatics and Gauss's Law notes, basic calculus
Tags: magnetic field, Faraday's Law, Lenz's Law, induced EMF, electromagnetic induction, RL circuit, LC circuit, time constant, angular frequency, torque on current loop, magnetic flux, inductor energy
This material bridges the gap between static electricity and time-varying fields. Once charges start moving (currents), they create magnetic fields, and changing magnetic fields create electric fields in return. That feedback loop is the engine behind generators, transformers, and every piece of electronics that uses an inductor. The RL and LC circuits here are the simplest versions of that interplay: how current grows in an inductor, and how energy sloshes between a capacitor and an inductor.
If you are comfortable with Gauss's Law and basic circuit analysis (Kirchhoff's rules, Ohm's Law), you are ready for this. The new conceptual layer is that changing flux drives induced currents, and inductors resist changes in current the way capacitors resist changes in voltage.
Faraday's Law says a changing magnetic flux through a loop induces an EMF, and Lenz's Law says the induced current opposes the change. RL circuits have exponential current growth with time constant tau = L/R. LC circuits oscillate sinusoidally at angular frequency omega = 1/sqrt(LC), with energy swapping between the capacitor's electric field and the inductor's magnetic field.
Magnetic flux (Phi_B)
The integral of the magnetic field over an area: Phi_B = integral of B dot dA. For a uniform field through a flat loop, Phi_B = BA cos(theta). Think of it as "how much magnetic field passes through the loop."
Faraday's Law of Induction
The induced EMF in a loop equals the negative rate of change of magnetic flux through the loop: EMF = -d(Phi_B)/dt. In simple terms, a changing magnetic environment around a wire loop pushes charges around the loop.
Lenz's Law
The direction of the induced current is such that it opposes the change in flux that produced it. Think of it as nature's resistance to change: if the flux is decreasing, the induced current flows in a direction that tries to maintain it.
Inductance (L)
A measure of how much magnetic flux a coil produces per unit current, in henries (H). An inductor with high inductance strongly resists rapid changes in current.
RL circuit time constant (tau = L/R)
The characteristic time for current to grow (or decay) in an RL circuit. After one time constant, the current reaches about 63% of its final value when growing, or drops to about 37% when decaying.
LC oscillation angular frequency (omega)
For an ideal LC circuit: omega = 1/sqrt(LC). This is the natural frequency at which energy oscillates between the capacitor and inductor, measured in rad/s. The ordinary frequency in Hz is f = omega/(2 pi).
Torque on a current loop
A current-carrying loop in a uniform magnetic field experiences a torque: tau = NIAB sin(alpha), where alpha is the angle between the magnetic moment and the field. N is the number of turns, I the current, A the area, B the field strength.
Energy stored in an inductor
U = (1/2)LI². The energy is stored in the magnetic field within and around the inductor, analogous to (1/2)CV² for a capacitor.
A rectangular conducting loop sits in the same plane as a long straight wire carrying constant current I. The wire's magnetic field decreases with distance: B = mu_0 I/(2 pi r).
When the loop moves away from the wire at constant velocity v:
The magnetic flux through the loop decreases, because the field is weaker farther from the wire.
By Lenz's Law, the induced current flows in the direction that tries to maintain the flux, which means it flows in the same sense as the current in the wire (on the near side). Working out the geometry with the right-hand rule: the induced current is counter-clockwise (if the wire's current runs upward and the loop is to the right).
The direction depends on the specific geometry, so on an exam you must apply Lenz's Law to the given setup rather than memorising a single answer.
How the induced EMF changes with distance:
The magnetic field from the wire falls off as 1/r. As the loop moves farther away, the rate of change of flux decreases (the field gradient flattens out). The induced EMF therefore decreases as the distance increases.
For a rectangular loop of dimensions L x W carrying current I in a uniform field B, the magnetic moment is m = ILW (for a single turn).
The torque is tau = m x B, with magnitude tau = ILWB sin(alpha).
The angle alpha here is the angle between the area vector (normal to the loop's plane) and the magnetic field. Be careful: if the problem states that "the plane of the loop makes an angle phi with the field lines," then the angle between the area vector and B is (90 degrees - phi). In that case, tau = ILWB sin(90 - phi) = ILWB cos(phi).
This distinction between "angle of the plane" and "angle of the normal" is a frequent source of lost marks.
An RL circuit (battery with EMF E, resistor R, inductor L in series) is switched on at t = 0.
At t = 0:
The inductor opposes any instantaneous change in current. Since the current was zero just before the switch closed, it remains zero at t = 0. The full battery voltage appears across the inductor.
So I(0) = 0.
As t approaches infinity:
The current reaches its steady-state value. The inductor behaves like a plain wire (no changing current, no induced EMF). The current is I_max = E/R.
Time constant:
tau = L/R. This is the time for the current to reach about 63% of E/R. After about 5 tau, the current is effectively at its maximum.
Current as a function of time:
I(t) = (E/R)(1 - e^(-Rt/L))
After a long time, the current is I = E/R. The energy stored in the inductor is:
U = (1/2)LI² = (1/2)L(E/R)²
This is a straightforward substitution, but exams like to ask you to "prove" or "show that" the expression holds, which means writing out both the steady-state current and the energy formula explicitly.
An LC circuit with L = 10 mH and C = 10 muF, capacitor initially charged to Q_0 = 100 muC:
Angular frequency:
omega = 1/sqrt(LC) = 1/sqrt(10 x 10⁻³ times 10 x 10⁻⁶) = 1/sqrt(10⁻⁷) = 1/(3.162 x 10⁻⁴) ≈ 3162 rad/s.
Rounding to a listed exam option: approximately 3162 rad/s (often written as 3,162 rad/s).
Maximum current:
Energy conservation: the maximum energy in the capacitor equals the maximum energy in the inductor.
(1/2)Q_0²/C = (1/2)LI_max²
I_max = Q_0/sqrt(LC) = Q_0 times omega = 100 x 10⁻⁶ times 3162 ≈ 0.316 A.
Alternatively, I_max = Q_0 times omega directly, since Q(t) = Q_0 cos(omega t) and I = -dQ/dt = Q_0 omega sin(omega t), so the amplitude of the current is Q_0 omega.
Faraday's Law: EMF = -d(Phi_B)/dt
Magnetic field from long straight wire: B = mu_0 I / (2 pi r)
Torque on a current loop: tau = NIAB sin(alpha)
RL circuit current (charging): I(t) = (E/R)(1 - e^(-t/tau)), tau = L/R
RL circuit current at t = 0: I = 0
RL circuit current at t = infinity: I = E/R
Energy in inductor: U = (1/2)LI²
LC angular frequency: omega = 1/sqrt(LC)
LC maximum current: I_max = Q_0 omega = Q_0/sqrt(LC)
LC charge: Q(t) = Q_0 cos(omega t)
LC current: I(t) = Q_0 omega sin(omega t)
Electromagnetic induction is how every electrical generator works: a coil rotates in a magnetic field (or a magnet rotates near a coil), the flux changes, and an EMF is induced. The RL time constant governs the switching behaviour of relay circuits and the current surges that occur when large motors are turned on. LC oscillations are the basis of every radio tuner: adjusting the capacitance or inductance changes the resonant frequency to select a particular station.
Students often think that the induced EMF in a loop near a current-carrying wire stays constant as the loop moves away. It does not; the EMF decreases because the field gradient weakens with distance.
A common mistake is to say the current in an RL circuit at t = 0 is E/R. That is the final (long-time) value. At t = 0, the inductor blocks any instantaneous current change, so I = 0.
Students sometimes confuse the RL time constant (tau = L/R) with the RC time constant (tau = RC). Notice the ratio is inverted: for RL, the inductance is in the numerator; for RC, the resistance is.
In LC circuits, students often forget that energy conservation links Q_0²/C to LI_max², and instead try to use Ohm's Law (which does not apply, since an ideal LC circuit has no resistance).
⚠️ Lenz's Law direction questions require careful use of the right-hand rule applied to the specific geometry. Do not guess based on memory from a different problem.
⚠️ The RL circuit at t = 0 (I = 0) and at t = infinity (I = E/R) are near-certain exam points. The time constant tau = L/R (not R/L) is easily confused with the RC version.
⚠️ LC oscillation problems almost always ask for omega, I_max, or both. Energy conservation is the fastest route to I_max.
⚠️ Torque on a current loop: read the angle definition very carefully. "Angle of the plane with respect to the field" and "angle of the normal with respect to the field" give different trig functions.
True or false: at t = 0 in an RL circuit with a battery, the current through the inductor is E/R.
Fill in the blank: the time constant of an RL circuit is tau = ______.
True or false: in an ideal LC circuit, the total energy oscillates between the capacitor and the inductor, but the sum is constant.
Fill in the blank: the angular frequency of oscillation in an LC circuit is omega = ______.
True or false: as a conducting loop moves away from a long straight current-carrying wire, the induced EMF increases.
Answers: 1. False (I = 0 at t = 0). 2. L/R. 3. True. 4. 1/sqrt(LC). 5. False (it decreases).
Q: In an RL circuit (battery EMF E, resistance R, inductance L), what is the current at t = 0 when the switch is first closed?
A: I = 0. The inductor opposes any instantaneous change in current from zero.
Q: What is the time constant for an RL circuit?
A: tau = L/R. Not R/L.
Q: An LC circuit has L = 10 mH and C = 10 muF. What is the angular frequency of oscillation?
A: omega = 1/sqrt(LC) = 1/sqrt(10⁻⁷) ≈ 3,162 rad/s.
Q: The capacitor in the LC circuit above is initially charged to 100 muC. What is the maximum current?
A: I_max = Q_0 omega = (100 x 10⁻⁶)(3162) ≈ 0.316 A.
Q: A rectangular loop carrying current I in a uniform magnetic field B has its plane making angle phi with the field lines. What is the torque magnitude?
A: tau = ILWB cos(phi). The angle between the area vector and B is (90 - phi), so sin(90 - phi) = cos(phi).
Q: Prove that the energy stored in an RL circuit inductor after a long time is U = (1/2)L(E/R)².
A: After a long time, the current reaches steady state: I = E/R. The energy in the inductor is U = (1/2)LI² = (1/2)L(E/R)².
The RL and LC circuits here are the building blocks of the RLC circuit covered in the AC circuits notes. Adding a resistor to an LC circuit introduces damping; driving it with an AC source introduces resonance. Faraday's Law is also one of the four Maxwell's equations, connecting this material to the electromagnetic waves covered in the EM waves and optics notes.
Faraday's Law, Lenz's Law, electromagnetic induction, magnetic flux, induced EMF, induced current, RL circuit, LR circuit, time constant, tau, L/R, exponential growth, inductor, inductance, henry, LC circuit, LC oscillation, angular frequency, omega, natural frequency, energy conservation, capacitor energy, inductor energy, torque on a current loop, magnetic moment, right-hand rule, steady-state current, transient response