Electromagnetic Induction

Magnetic flux through a loop is Φ_B=∫B·dA. For a flat loop in uniform B, Φ_B=BA cosθ, with θ measured from the area normal. Flux changes if field strength, enclosed area, or orientation changes. A moving loop entirely inside a uniform field can have constant flux and zero net induced emf.

Faraday’s law gives ε=−N dΦ_B/dt for N identical linked turns. The minus sign expresses opposition to change, not an automatically negative measured voltage. Choose a loop direction and the associated normal before interpreting signed flux.

If outward flux increases, an induced current produces an inward field. If outward flux decreases, the induced field is outward to resist the decrease. Use the right-hand grip rule to turn that field direction into a current direction. The induced field need not cancel the original field completely.

A loop crossing a finite field region has different phases: entering changes overlap area, being fully inside can keep flux constant, and leaving reverses the flux change. Track the signed flux as a function of position and then time; this is more reliable than memorizing one current direction.

For a conducting rod of length ℓ moving perpendicular to uniform B and itself at speed v, charge separation produces magnitude ε=Bℓv. With a completed circuit of resistance R, I=BℓvR. The current-carrying rod experiences magnetic drag F=B²ℓ²vR.

To maintain constant speed, an external force supplies power Fv=I²R. The electrical energy is supplied by mechanical work. If the rod slows freely, kinetic energy decreases while thermal energy increases. Omitting the induced force would allow energy to appear without a source.

A coil’s flux linkage is proportional to current in a linear model: NΦ=LI. Its induced emf is −L dI/dt. A current cannot jump instantaneously through an ideal inductor with finite voltage. Unlike a resistor, an ideal inductor does not continually dissipate power; it stores magnetic energy U=LI²2.

For a series RL circuit connected to a source, ε−IR−L dI/dt=0. Starting at zero, I=(εR)(1−e^−tR/L). The time constant is LR. Immediately after closing the switch, current is zero; long after, the ideal inductor has zero voltage drop while the resistor carries εR.

When the battery is removed and the inductor discharges through a resistor, I=I₀e^−tR/L. The inductor’s polarity reverses so that current continues in its original direction while decreasing. The resistor converts stored magnetic energy into thermal energy.

The relevant resistance is the entire resistance in the actual discharge path. A switch diagram must show that path; simply removing the source without a closed path can produce a large voltage rather than the assumed gentle exponential. Check both continuity at t=0 and the long-time state.

For an ideal isolated LC circuit, L d²qdt²+qC=0. The charge oscillates with angular frequency ω=1√(LC); current is dq/dt and is one quarter-cycle out of phase. Total energy q²(2C)+LI²2 remains constant when resistance is neglected.

At maximum capacitor charge, current and magnetic energy are zero. When capacitor charge is zero, current magnitude and magnetic energy are largest. A real circuit’s resistance damps the oscillation, so the ideal solution is a model with a specified approximation rather than a claim that energy losses never occur.

Magnetic flux through a surface is the integral of B·dA. For a uniform field and flat loop it becomes BA cos θ, where θ is between B and the area normal. If the angle is given relative to the plane, convert it. A field parallel to the plane gives zero flux through that plane.

Flux can change because field magnitude, loop area or orientation changes. A conducting loop moving through a nonuniform region can experience changing flux even if the source field is steady in time. Conversely, a loop moving entirely within a uniform field at fixed area and orientation can have constant flux.

Flux is a signed quantity defined by the chosen normal. A consistent normal and loop orientation are essential when assigning an emf sign. The observable current direction must ultimately agree with the physical opposition described by Lenz’s law.

Faraday’s law relates induced emf to the rate of change of linked flux, with the number of turns included for a coil. A large flux does not necessarily produce an emf if it is constant. A small flux changing rapidly can produce a substantial emf.

First determine whether the original flux in a chosen direction is increasing or decreasing. Then choose an induced field that opposes that change, and finally use the right-hand rule to find the current direction. The induced field does not always oppose the original field: it reinforces an original field that is decreasing.

Emf is not automatically current. Current additionally depends on a closed conducting path and its electrical properties. An open loop can have an induced emf without the same circulating current that would flow through a closed low-resistance loop.

For a conducting rod moving appropriately through a uniform field, magnetic forces separate charges until an electric field develops. In the standard perpendicular geometry, emf magnitude is BLv. This expression requires the relevant perpendicular components and should not be applied unchanged to every orientation.

If the rod completes a circuit, the resulting current can experience a magnetic force opposing the driven motion. An external agent must supply work to maintain constant speed. In an ideal resistive circuit, mechanical power can equal the electrical heating rate, demonstrating energy conservation.

Lenz’s law prevents a self-amplifying process that would generate useful electrical energy without an input. When a magnet slows near a conducting region, kinetic energy can become electrical and thermal energy. Explain the transfer rather than treating the magnetic field as an unlimited fuel source.

A changing current produces a changing magnetic field and a self-induced emf. An ideal inductor opposes changes in current through its L di/dt relation; it does not simply oppose any current forever. In an ordinary ideal finite-voltage circuit, inductor current cannot jump instantaneously.

For a simple RL step response, LR sets the time constant. Initially, a previously unenergized inductor prevents an instantaneous current jump; at long time under DC, an ideal inductor’s voltage approaches zero while current can remain nonzero. This is complementary to the long-time behavior of an ideal capacitor.

Stored magnetic energy is ½LI². When a source is removed, that energy can sustain current temporarily and be dissipated in a resistor. A switching analysis must use the initial inductor current, the new circuit topology and the final state. Confusing current continuity with voltage continuity swaps the roles of inductors and capacitors.

A conducting rod of length ℓ moving at speed v perpendicular to a uniform B experiences charge separation through magnetic force. In the standard perpendicular geometry, the motional emf magnitude is Bℓv. If it forms a closed circuit of resistance R, current magnitude is BℓvR under the ideal assumptions.

The induced current produces a magnetic force opposing the motion that changes flux. An external agent maintaining constant speed must supply mechanical power. Using F=IℓB gives Fv=I(Bℓv)=I²R, matching resistive heating. This energy check connects Lenz’s law with conservation rather than treating its direction rule as arbitrary.

An open rod can have an emf without a sustained closed-loop current. A changing magnetic flux can also induce emf in a stationary loop. Do not assume that all induction requires a moving conductor, or that the presence of emf alone guarantees a particular current without knowing the circuit.

An inductor’s self-induced emf opposes change in current: the sign follows −L di/dt under the usual convention. Its stored magnetic energy is LI²2. A large current is not necessarily a large induced emf; an unchanging current has di/dt=0 in the ideal inductor.

For a series RL circuit connected to a constant source, Kirchhoff’s rule gives V−IR−L di/dt=0. Starting from zero current, I=(VR)(1−e^−tR/L); the time constant is LR. At long times, an ideal inductor has zero voltage drop while the resistor limits current.

When the source is removed but a resistive discharge path remains, current decays exponentially and stored magnetic energy becomes thermal energy. Current cannot jump instantaneously in the ordinary finite-voltage ideal model. Opening a path abruptly can produce a large voltage, showing why a complete switching circuit matters to the physical prediction.