U5: Electromagnetism

B is the magnetic field, and dA is the infinitesimal area, pointing outwards. The unit for magnetic flux is the “weber” (Wb), in honour of Wilhelm Weber. 1 Wb = 1 Tm2.

Magnetic Flux Over A Closed Area

Magnetic flux over a closed area is always zero.

∮𝐵 * 𝑑𝐴 = 0

Because magnetic field lines only go in a loop, there’s always an equal amount of flux flowing out as there is entering the surface.

Changing Magnetic Flux

Changes to magnetic flux occur for a number of reasons:

The magnetic field is changing. Perhaps the field is generated by a time-dependent source.
The orientation is changing. This could be because the surface area is moving relative to the magnetic field.
The area is changing. The surface area from which the flux is calculated is changing.

When magnetic flux is changing, and there’s a wire in the magnetic field, an electromotive force (emf, or 𝜀) is created in the wire. Unlike in a normal circuit, where emf is concentrated at the terminals of a battery, this induced emf is spread across the whole wire.

emf is work per unit charge, which means that there’s an electric field inside the wire that’s moving the charges.

Here, there’s a magnetic field B going into the page, and the direction of electric field E corresponds to an increase in magnetic flux.

Faraday’s Law

Faraday’s law states that the rate of change of magnetic flux produces an electromotive force.

𝜀 = ∮ 𝐸 * 𝑑𝑙 = − 𝑑𝛷/𝑑𝑡

AC Generators

Let’s apply Faraday’s law. An AC generator makes use of a coil rotating against a fixed magnetic field, which changes the flux. Let those magnets generate a uniform magnetic field B, with a coil of N turns and an area A, rotating at an angular frequency of 𝜔. This uses a little C: Mechanics knowledge, so if you don’t know about angular frequency yet, don’t sweat it.

The angle between the coil and the field is:

𝜃 = 𝜔𝑡 + 𝛿

(𝛿 is the initial angle.) The magnetic flux through the coil is time-dependent, and given by:

𝛷 = 𝑁𝐵𝐴𝑐𝑜𝑠 𝑐𝑜𝑠 (𝜃) = 𝑁𝐵𝐴𝑐𝑜𝑠 𝑐𝑜𝑠 (𝜔𝑡 + 𝛿)

𝜀 = −𝑑𝜑𝑚/𝑑𝑡 = 𝑁𝐵𝐴𝜔 𝑠𝑖𝑛 𝑠𝑖𝑛 (𝜔𝑡 + 𝛿)

𝜀 = 𝜀max 𝑠𝑖𝑛 𝑠𝑖𝑛 (𝜔𝑡+ 𝛿) , 𝑤ℎ𝑒𝑟𝑒 𝜀max = 𝑁𝐵𝐴𝜔

When you slide that rod to the right with a speed v, you are changing the flux through the loop created by the u-wire and the rod.

𝜑 = 𝐵𝐴 = 𝐵𝑙𝑥

𝑑𝜑/𝑑𝑡 = 𝐵𝑙 * (𝑑𝑥/𝑑𝑡) = 𝐵𝑙𝑣 = 𝜀

But remember that the “motional emf” produced is spread over the entire circuit.

Lenz’s Law

Just looking at the last situation, something interesting happens when the current runs through the wire.

There’s an “induced magnetic field” coming out of the page, opposite to the field that generated the current.

Lenz’s Law:

The induced emf and the induced current are in such directions as to oppose the change that produces them

It’s basically the conservation of energy.

Inductance

Back emf

Consider this circuit with a voltage source, coil and switch.

When the switch is closed and current begins to flow, the coil begins to generate a magnetic flux inside. As the current changes (increasing with time, initially), it self-induces a “back emf” that opposes the change in current. A current can’t jump from zero to some value instantaneously, obviously.

When you break the circuit, you cause very fast magnetic flux change, which creates a large induced back emf, proportional to the rate of change of magnetic flux, which creates a large voltage drop across the switch. Large voltage across large metal contacts creates a very strong electric field, strong enough to tear electrons off of air molecules (dielectric breakdown), which creates a spark.

Self-Inductance

A solenoid carrying a current generates a magnetic field, given by the Biot-Savart law (or Ampere’s law, too):

𝐵 = 𝜇0𝑁𝐼/𝐿

B is proportional to I, so the magnetic flux through the solenoid is also proportional to I.

𝛷M = 𝐿𝐼

L is the self-inductance of the coil. In a solenoid, self-inductance is given by

𝐿 = 𝛷𝑀/𝐼 = 𝜇0𝑛2𝐴𝑙

n here is the number of coil turns per unit length, A is the cross-sectional area and l is the length of the solenoid. (Al is the enclosed volume.)

If the current changes, magnetic flux changes, too, which means that an electromotive force is induced in the circuit. According to Faraday’s Law, once again:

𝜀 = −𝑑𝛷𝑚/𝑑𝑡 = −𝐿(𝑑𝐼/𝑑𝑡)

The self-induced emf is proportional to the rate of change of current.

Magnetic Energy

A capacitor stores energy in its electric field, and an inductor coil carrying a current I stores energy in its magnetic field.

𝑈𝑀 = (1/2) * 𝐿𝐼2

There’s also something called the magnetic energy density:

𝜂𝑀 = (𝐵2)/(2𝜇0)

Maxwell’s Equations

Let’s cap off this unit (and exam review) with 4 equations that embody the subject of electromagnetism.

Gauss’s Law

This is useful for calculating electric fields, especially when there’s symmetry involved.

𝛷𝐸 = ∮ 𝐸 * 𝑑𝐴 = 𝑞in/𝜖0

It can also be shown this way:

∮ 𝐸⋅ 𝑑𝐴 = 𝑄encl/𝜖0

Gauss’s Law For Magnetic Fields (Closed Surfaces)

Here, we have our statement that magnetic field lines always form closed loops that encircle the current that generates them.

∮ 𝐵 * 𝑑𝐴 = 0

Basically, if you have a closed surface, there’s always as much flux out as there is entering, due to the nature of magnetic field lines.

Faraday’s Law

This is a very important law as well: it shows that changing magnetic flux induces emf, and an electric field.

𝜀 = ∮ 𝐸 * 𝑑𝑙 = −𝑑𝛷𝐵/𝑑𝑡

Ampere-Maxwell Law

We start this one off with Ampere’s law, which is useful for calculating magnetic fields.

∮ 𝐵 * 𝑑𝑙 = 𝜇0𝐼C

Maxwell added on a little bit to the end, which shows that a changing electric field produces a magnetic field:

∮ 𝐵 * 𝑑𝑙 = 𝜇0𝐼 +𝜇0𝜖0 * (𝑑𝛷E/𝑑t)

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