Honors Physics Unit 20 Review: Magnetism

A magnetic field is a vector field that describes the magnetic influence at each point around a magnet or moving charge. It is measured in tesla (T). Every magnet has a north and south pole, and cutting a magnet in half just gives you two smaller magnets.

Field lines leave the north pole, curve around outside the magnet, enter the south pole, and continue through the magnet to form closed loops. Where lines bunch together, the field is stronger. Around a straight wire, the lines are circles. Point your right thumb along the current and your fingers curl in the direction of the field.

The magnetic force magnitude on a moving charge is $F = |q|vB\sin\theta$, where $\theta$ is the angle between $v$ and $B$. For a straight wire in a uniform field, the force magnitude is $F = ILB\sin\theta$, where $\theta$ is the angle between the current and the field. For direction, point your fingers along $v$ (or $I$), curl them toward $B$, and your thumb gives the force on a positive charge. Flip it for a negative charge. If magnetic force is the only force and the charge moves perpendicular to a uniform field, it follows a circle with $r = \frac{mv}{|q|B}$. A velocity component parallel to the field produces a helical path.

Worked example. A proton ($m = 1.67 \times 10^{-27}$ kg, $q = 1.6 \times 10^{-19}$ C) moves at $2.0 \times 10^6$ m/s perpendicular to a 0.50 T field. Then $r = \frac{(1.67 \times 10^{-27})(2.0 \times 10^6)}{(1.6 \times 10^{-19})(0.50)} = \frac{3.34 \times 10^{-21}}{8.0 \times 10^{-20}} \approx 0.042$ m, or about 4.2 cm.

A motor turns electrical energy into motion. Current runs through a coil sitting in a magnetic field. The forces on opposite sides of the coil point in opposite directions, which creates a torque that spins it. In a DC motor, a split-ring commutator reverses the current every half turn so the coil keeps rotating the same way.

A generator does the reverse. A turbine or crank spins a coil in a magnetic field, the flux through the coil keeps changing, and an emf is induced. A spinning coil produces an alternating emf, which can drive alternating current (AC) in a closed circuit. Power plants commonly generate AC.

A transformer changes AC voltage using two coils on a shared iron core. The changing current in the primary makes a changing flux, which induces a voltage in the secondary. For an ideal transformer, the voltage ratio equals the turns ratio: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$. It also conserves power, so $V_pI_p = V_sI_s$. A transformer needs changing flux, normally supplied by AC. Steady DC does not induce a sustained secondary voltage, though switching it on or off can induce a transient voltage.

Worked example. An ideal transformer has 400 turns on the primary and 20 on the secondary, with 120 V in. Then $V_s = 120 \times \frac{20}{400} = 6.0$ V. If the secondary delivers 2.0 A, power is $6.0 \times 2.0 = 12$ W, so the primary draws $\frac{12}{120} = 0.10$ A.

Power companies step voltage up to hundreds of kilovolts for transmission. For the same transmitted power, higher voltage means lower current. Wire losses follow $P = I^2R$, so cutting current by 10 cuts losses by 100 if the wire resistance stays the same.

A changing magnetic flux through a loop induces an emf. A steady field through a still loop does nothing. Magnetic flux measures how much field passes through an area. For a uniform field through a flat loop, $\Phi_B = BA\cos\theta$, where $\theta$ is the angle between $B$ and the line perpendicular to the loop. Its unit is the weber (Wb).

You can change flux three ways. You can change $B$, change the area $A$, or change the angle $\theta$ by rotating the loop. Faraday's law gives the magnitude of the average emf over an interval, $|\varepsilon_{\text{avg}}| = N\frac{|\Delta\Phi_B|}{\Delta t}$. For a closed circuit with resistance $R$, Ohm's law gives the current magnitude, $I = \frac{|\varepsilon|}{R}$. A rod of length $L$ moving at speed $v$ has a motional emf of $\varepsilon = BLv$ when the rod, its velocity, and the uniform magnetic field are mutually perpendicular.

Lenz's law gives the direction. In a closed circuit, the induced current makes a field that opposes the change in flux. If an external field through the loop has a fixed direction and gets stronger, the induced field points opposite it. If that field gets weaker, the induced field points the same way. Then use the right-hand rule to find which way the current circulates. This is conservation of energy at work, since you must do work to push a magnet into a coil.

Worked example. A 50-turn coil of area 0.010 m² sits perpendicular to a field that rises from 0 to 0.40 T in 0.20 s. The flux per turn changes by $(0.40)(0.010) = 0.0040$ Wb. So $\varepsilon = 50 \times \frac{0.0040}{0.20} = 1.0$ V. With a resistance of 5.0 Ω, $I = \frac{1.0}{5.0} = 0.20$ A.