U4: Magnetism

What Is Magnetism?

Magnetism is generated by moving charged particles: it can be either single moving particles, or electric current moving through a conductor. It can also be generated by permanent magnets, or the Earth.

Magnetism affects other moving charged particles. Like an electric field, it can be represented with a vector field, called a magnetic field (big surprise!). The magnetic field has units called tesla, and magnetic field lines have no ends: however, unlike electric field lines, which run to infinity, magnetic field lines always run in loops.

Magnetic Field Generated By A Moving Point Charge

A moving point charge q moving with velocity v has a magnetic field given by the following equation:

𝐵 = (𝜇0/4𝜋) * (𝑞𝑣 * 𝑟̂)/(𝑟2)

B is the magnetic field in units of tesla (T), q is the charge, v is the velocity of the charge, r is the distance from the moving charge, 𝑟̂is the radial outward unit vector from the charge (for direction), and 𝜇0 is the permeability of free space, which is a measure of how well a space can become magnetized.

𝜇0 is about 4𝜋 * 10-7𝑇𝑚/𝐴.

Magnetic Field Generated By A Current

The magnetic field generated by an electric current is really the sum of all the contributions of each charge moving along through the conductor. The total magnetic field in the wire is the integral of the contributions of the current in each infinitesimally small section of the conductor. We use the Biot-Savart Law for this.

𝑑𝐵 = (𝜇0/4𝜋) * (𝐼𝑑𝐿 * 𝑟̂)/(𝑟2)

Magnetic Field In An Infinitely Long Wire

Given an infinitely long and straight wire, the Biot-Savart law simplifies greatly. Current is given by:

𝐵 = 𝜇0(𝐼 × 𝑟̂)/2𝜋𝑟 𝑜𝑟 𝐵 = 𝜇0𝐼/2𝜋𝑟

The equation on the right is the scalar form, for when you don’t want or need to deal with vectors and directions.

Magnetic Field With Solenoids

The magnetic field inside of a solenoid is uniform, and the strength is given by:

𝐵 = 𝜇𝑁𝐼/𝐿

Here, L is the length of the solenoid, while N is the amount of coils in the solenoid. The direction of B is determined using the right hand rule.

Ampere’s Law

Gauss’s Law is used to calculate electric fields and fluxes, and Ampere’s Law does the same for magnetic fields for symmetric configurations.

Ampere’s Law using a closed curve C around a current, called an Amperian Loop. We do a closed line integral instead of a closed surface integral on the curve.

∮C  𝐵 * 𝑑𝑙 = 𝜇0𝐼c

d𝑙 is an infinitesimal length along the closed curve. Ic is the net current that penetrates the area that is bound by C.

How Do We Apply It?

Remember that infinitely long wire from before? Well, we know that it must generate a magnetic field that only depends on radial distance, due to symmetry. We can draw an Amperian loop that is just a circle of radius r around the wire, and do some calculations.

∮c  𝐵 * 𝑑𝑙 = 𝜇0𝐼c ⟶ 𝐵(2𝜋𝑟) = 𝜇0𝐼 ⟶ 𝐵 = 𝜇0𝐼/2𝜋𝑟

Lorentz Force Law

Magnetic fields are generated by moving charges. A moving charge will also create an electric field. So, if you want to figure out the influence on another moving charge affected by both E and B, you add their influences together, which gives the Lorentz force law.

𝐹 = 𝑞(𝐸 + 𝑣 * 𝐵)

Force On A Current-Carrying Conductor In A Magnetic Field

B generated by a moving charge or conductor carrying a current will exert a force on another conductor that carries a current.

𝐹M = 𝐼𝑙 * 𝐵

Here, I is treated as a vector. 𝑙 is the length of the conductor, in meters.

2 Wires Exerting Force Together

When we have 2 parallel current-carrying wires of length L, currents I1 and I2, and distance between them of r, the force of the magnetic field B from I1 on I2 is:

𝐹 = 𝐼2𝐿𝐵 = 𝜇0𝐼1𝐼2𝐿/2𝜋𝑟 ⟶ 𝐹/𝐿 = 𝜇0𝐼1𝐼2/2𝜋𝑟

I1 and I2 are pulled together because of Newton’s Third Law of motion.

Circular Motion In Magnetic Fields

When a charged particle enters a magnetic field at a right angle, the magnetic force FM is perpendicular to both velocity v and the magnetic field B, because it’s a cross product. This will result in circular motion.

The centripetal force FC is provided by the magnetic force. Let’s equate the 2 expressions.

𝑚𝑣2/𝑟 = 𝑞𝑣𝐵

We can now rearrange this equation to solve for the radius of motion of a charge with known mass, or work out the mass of the charge using its radius.

𝑟 = 𝑚𝑣/𝑞𝐵 𝑎𝑛𝑑 𝑚 = 𝑞𝑟𝐵/v