Electromagnetic Induction, Lenz’s Law, and Electric Power Applications

Electric fields come from two sources:
- Charge separation (electrostatic, e.g., batteries), with fixed EMF magnitude and sign.
- Changing magnetic flux (Faraday, 1831), which induces currents.
Experimental setup: A conducting loop with an ammeter, no battery. An infinite solenoid passes through the loop, creating a uniform magnetic field B out of the page. Magnetic flux (Phi_B) through the loop is B times A.
Observations:
- Steady field: No current.
- Increasing B: Clockwise current.
- Decreasing B: Counter-clockwise current.
Equivalent magnet experiment: Moving a bar magnet toward a loop induces current one way; moving it away induces current the opposite way. A static magnet induces no current.

Faraday's Law: A time-varying magnetic flux through a conductor induces an EMF. For a single loop, the magnitude of EMF (E) is the absolute value of the rate of change of magnetic flux (d(Phi_B)/dt).
For a tilted loop in a uniform field, magnetic flux (Phi_B) is B times A times cos(phi), where phi is the angle between B and A.
Three ways to change magnetic flux (d(Phi_B)/dt not zero):
1. Changing field magnitude (B).
2. Changing area (A).
3. Changing orientation angle (phi).
For a coil with N turns, total EMF (E) is N times the absolute value of d(PhiB)/dt. The product N times PhiB is called flux linkage.

Lenz's Law determines the direction of induced EMF/current. It states: 'Induced current creates a magnetic field that opposes the change in magnetic flux that caused the current'.
Right-hand rule for direction: Point your thumb opposite to the change in B-field; your fingers show the current direction.
Another right-hand rule: Fingers wrap in current direction, thumb points to loop's B-field.
- When area changes: Shrinking area means flux decreases, so loop's B-field supports the external B-field. Growing area means flux increases, so loop's B-field opposes the external B-field.
- When angle changes (rotation): As flux decreases from maximum, loop's B-field aligns with external B-field. As flux increases from zero, loop's B-field is anti-parallel to external B-field.
Faraday's Law with Lenz's direction: EMF (E) = -N times d(Phi_B)/dt (The negative sign indicates opposition).

Lenz's Law ensures energy conservation: Induced currents oppose flux change, converting mechanical work into electrical and thermal energy, preventing perpetual energy gain.
Pushing a magnet toward a coil causes repulsion, requiring work. This work is converted to electrical energy (and heat) in the coil. If Lenz's law were reversed, it would violate energy conservation.

Slide Generator Components: U-shaped rails, a movable conducting bar of length l sliding right at velocity v, and a uniform constant magnetic field B into the page.
As the bar moves, the loop area (l times x(t)) grows, increasing flux and inducing a clockwise current (lighting a bulb). Motional EMF (E) is derived from magnetic force on charges (F_B = qvB), leading to E = B times l times v. Potential difference across the bar is V = E times l = B times l times v.
Mechanical input (power) balances electrical output plus magnetic damping (back-force).

Eddy Currents: Circulating currents induced in bulk conductors moving through non-uniform magnetic fields. They obey Lenz's Law and create magnetic drag (like friction).
- Applications: Magnetic braking (trains, instruments), converting kinetic energy to heat without wear.
- Drawbacks: Heat loss in generators; minimized by laminations.

AC Generator: A rotating coil between magnet poles connects to an external circuit via slip rings and brushes.
- Increasing flux: Negative EMF (current one way).
- Flux at maximum: EMF is zero.
- Decreasing flux: Positive EMF (current opposite way).
One-cycle analysis:
- Flux: PhiB = BA cos(omega t) = Phimax cos(omega t).
- Induced EMF (E): Emax sin(omega t), where Emax = NBA omega.
- Current (I): I_max sin(omega t).

AC Motor: An AC current produces torque, resulting in mechanical rotation. Developed by Tesla, crucial for AC adoption.

Faraday's Disk: A copper disk rotating in a uniform B-field. Charges are pushed radially, producing DC current from axle to rim. It was the first EM generator (1831), but had low power output.

Gramme Dynamo & Modern DC Machines: A coil rotates in a B-field, with output directed by a split-ring commutator to maintain constant current direction in the external circuit.
- AC generators use two slip rings; DC generators use a single split ring.
- The commutator reverses connections when the coil flips, keeping external polarity unchanged.

Time-varying DC Output: Magnitude pulsates, direction is fixed.
- For a single loop, EMF (E) = Emax times absolute value of sin(omega t). Current (I) = Imax times absolute value of sin(omega t).
- Multiple coils phased 90° apart reduce ripple, producing nearly steady DC, a refinement used by Edison.

Historical context: 'War of the Currents' between Edison (low-voltage DC) and Tesla/Westinghouse (high-voltage AC). Key issue: power loss (P_loss = I^2 R) in long wires.

RMS (Root Mean Square) quantities for AC:
- RMS EMF (Erms) = Emax / sqrt(2).
- RMS Current (Irms) = Imax / sqrt(2).
- Average power (Pavg) for a resistive load = Irms times E_rms.
- U.S. mains standard: Erms = 120V (Emax approx 170V).

Power-loss example:
- DC case (240V, 150kW, 0.25 Ohm resistance): Current is 625A, power loss is 98kW (over 65%!).
- AC case (24kV, 150kW, 0.25 Ohm resistance via step-up transformer): Current is 6.25A, power loss is 9.8W (negligible). This demonstrates AC's efficiency for long-distance transmission.

Transformers: Work only with AC (due to changing flux).
- Ideal voltage ratio: Secondary voltage (Vs) / Primary voltage (Vp) = Secondary turns (Ns) / Primary turns (Np).
- Ideal power (no loss): Primary power = Secondary power, meaning Is Vs = Ip Vp, and Is / Ip = Np / Ns.
- Step-up transformers (Ns > Np) increase voltage (Vs > Vp) and decrease current (Is < Ip), beneficial for power transmission.

Outcome of the 'War of the Currents': The 1893 Chicago World's Fair and the 1895 Niagara Falls plant proved AC's effectiveness for long-distance power transmission. The modern grid uses high-voltage AC transmission with local step-down transformers for distribution, cementing AC's dominance, though HVDC links exist for specific uses.