Magnetism

Magnetic fields are caused by the spinning of individual electrons. In objects that do not exhibit any magnetic properties, the spin of the electrons is random so there is no net magnetic field. In objects that do exhibit magnetic properties, the electrons’ spins have been aligned with each other so that they create a net magnetic field. Magnetic fields are similar to electric fields in that they interact with charged objects, but it is important to remember that magnetic fields only interact with charged objects when they’re moving perpendicular to the direction of the field (if it’s not moving perpendicular, use the component of the object’s motion that is perpendicular).

**Electromagnetism: **The relationship between electricity and magnetism - the two come hand in hand, since electric currents produce magnetic fields!

**Magnetic Field: **The region surrounding a magnetic source in which the magnet has a detectable magnetic force. Magnetic materials and electrical currents both can create magnetic fields. SI units: T

**Magnetic Field Lines: **Invisible lines that indicate the magnetic field of an area. They have a direction and point away from the north pole towards the south pole. Similar to electric fields, the density of the magnetic field lines indicates the strength of the field. In the picture below, we can see the magnetic field is strongest near the poles of the magnet.

Image Credit: Mirek2, CC-BY-SA 3.0

**Magnetic Flux: **Measures how much magnetic field passes perpendicularly through a given area. We can think of it as how many magnetic field lines pass through the region. SI units: Wb.

**Solenoid: **A coil of wires with turns that generates a uniform magnetic field.

**Electromagnetic Induction: **When a current is induced in a conductor moving through a magnetic field. To induce a current in a wire, we must change the magnetic fiux. To do so, we can change the magnetic field, change the wire’s orientation/area, or move the wire out of the magnetic field.

**Inductance: **The resistance of a wire to a change in current. SI units: H

**Self Inductance: **When the current in a wire changes, it creates a back emf, an induced voltage that opposes any change in current.

**Mutual Inductance: **The current in one coil of wire affects the current in another (induces an emf).

**Inductor: **Inductors resist changes in current by creating a back emf. They are usually no more than a coil of wires that wrap around a magnetic core.

**Faraday’s Law: **The induced emf is proportional to the rate of change of the magnetic flux.

**Lenz’s Law: **The induced current will always flow in the direction that generates a magnetic field to oppose the change in flux.

**RL Circuits: **Circuit made up of a resistor, inductor, and voltage source.

**LC Circuits: **Circuit made up of an inductor and a fully charged capacitor.

**RLC Circuits: **Circuit made up of an inductor, resistor, and fully charged capacitor.

**Hall Effect: **An effect observed when current passes through a conductor perpendicular to a magnetic field. The magnetic field will exert a force perpendicular to the direction of the current, pushing the electrons to one side of the conductor and creating a potential difference across the two sides of the conductor.

Relativity cont.

\(B = \frac{\mu_0 I}{2\pi^{l}}\)

B - magnetic field at a distance r away from the wire

\(μ_0\) - permeability of free space \((= 4π × 10^{-7} \frac{T . m}{A})\)

*First Right Hand Rule:* To find the direction of the magnetic field around a wire, point your thumb in the direction of the current and curl your fingers - that is the direction of the magnetic field!

Image Credit: Rnkv2, Public Domain

In **solenoids**, the magnetic field is uniform and contained within the coils.

\(B =\mu_n nI\)

μ0 - permeability of free space \((= 4π × 10^{-7} \frac{T . m}{A})\)

n - number of coils I - current

In solenoids, the magnetic field is uniform and contained within the coils. To find the direction of the field, curl the fingers of your right hand in the direction of the current, and your thumb will point in the direction of the magnetic field!

Image Credit: P.wormer, CC-BY-SA 3.0

\(F = qvB sin θ\)

q - particle’s charge v - particle’s velocity

θ - angle between particle and magnetic field’s directions

*Second Right Hand Rule*: To find the direction of the force from the magnetic field, point your index finger in the direction of the velocity, your middle finger in the direction of the magnetic field, and your thumb will point in the direction of the force!

Image Credit: Acdxr, GNU-FDL 1.2

F = BIL sin θ

I - current in wire, L - length of wire, θ - angle between wire and magnetic field

We can use the second right hand rule to find the direction of the force by remembering that a current is a collection of charges all moving in the same direction!

**Inductance **opposes change in current. In **self inductance**, when the current in a circuit changes, there will be an in-duced emf called **back emf **that opposes this change (**Lenz’s law**). Similarly, if the magnetic field changes, **magnetic flux** will change, which will result in an induced current that opposes this change in flux.

Faraday’s law: \(\epsilon = \frac{\Delta \phi}{\Delta t}\)

Because of this equation, we know that whenever there is a change in flux, there is an induced emf.

Magnetism problem guide

In circuits, **inductors **are represented by:

We use the symbol

to represent where the magnetic field comes out of the page and the symbol

to represent where the magnetic field goes into

the page. We can use these symbols for all vector quantities (not just for magnetic fields, but also for other vectors such as forces and velocities).

In **RL circuits**, when the switch is first closed, the inductor resists any change in current created by the voltage source by creating a back emf. Thus, it takes a longer time for the current to reach its equilibrium value, increasing at an exponential rate before reaching a steady rate. On the other hand, the voltage across the inductor is largest when the switch is first closed and decreases exponentially towards zero.

If the battery is removed, rather than the current immediately becoming zero, the inductor will resist the change in current and the current will take some time before reaching zero.

**LC circuits** are unique in that they have an oscillating current, charge, and voltage. When the switch is closed, the charged capacitor causes a current to ow until the capacitor is fully discharged. Remember that since there is an inductor, this discharge takes longer than if there was no inductor because the inductor

because the inductor opposes any change in current. Once the capacitor is fully discharged, the inductor, resisting any change in current flow, recharges the capacitor. When the capacitor is fully charged again, the cycle repeats.

An **RLC circuit**, like a LC circuit, oscillates between charging and discharging the capacitor. However, the presence of the resistor causes the current to decrease each time it runs through the resistor, resulting in a damped oscillation.

Faraday’s law

\(\epsilon= \frac{\Delta \phi}{\Delta t}\)

ε - emf

Φ - magnetic flux

t - time

For solenoid:

\(\phi = NB . A\)

N - number of loops of wire

**B **- magnetic field

**A** - area inside the loop

For inductor:

\(E = \frac{1}{2}LI^2\)

E - energy stored in inductor

L - inductance

I - current

\(\epsilon = L \frac{\Delta I}{\Delta t}\)

There are two long wires of length l containing currents 1 and 2. If the distance between them is r, what is Fb, the force they exert on each other?

Magnetism

\(B_1 = \frac{\mu_{\circ}I_1}{2 \pi r}\)

the magnetic force on wire 2 due to wire 1

\(F_{b2} = I_2IB_1\)

the force on wire 1 due to \(B_1\)

\(F_{b2} = \frac{\mu_{\circ}I_1I_2l}{2\pi r}\)

by substitution, we find the force on wire 2

To find the direction of \(F_{b2}\), we use the first right hand rule. We point our thumb in the direction of current 1 (to the right) and curl our fingers to find that the magnetic field of wire 1 points into the screen at wire 2. Using the second right hand rule, we point our index finger in the direction of wire 2's current (to the right), our middle finger in the direction of the magnetic field (into the screen), to find that our thumb (the force felt on wire 2) is in the upwards direction.

Because of Newton's Third Law, we know that the forces felt by the two wires are equal and opposite. Therefore the force on wire 2 is equal to the force on wire 1 and in the opposite direction. In other words, since the force on wire 2 is upwards, the force on wire 1 is downwards, and they both equal \(F_{b2} = \frac{\mu_{circ}I_1I_2l}{2\pi r}\)

You are given the setup shown below. The rod is a conductor and is slid slowly to the right. What is the value of the induced emf?

\(\phi = NB . A\)

\(A = Ix\)

x is the horizontal length of the circuit

\(\phi = Blx\)

because N = 1

\(c = \frac{\Delta \phi}{\Delta t}\)

\(\epsilon = \frac{Blx}{t}\)

x/t is equivalent to velocity v

\(\epsilon = Blv\)