Magnetism

## Big Picture

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).

## Key Terms

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

Both magnetic and electric fields store energy and  can  be  thought  of  as  vector  force  fields that move particles in specific directions.

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.

## Calculating Magnetic Fields

### Magnetic Field of a Wire

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

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!

Test Taking Tip: It may seem obvious, but remember to use your right hand! If you use your right hand to write, it is easy to forget and accidentally use your left hand.

Image Credit: Rnkv2, Public Domain

### Magnetic Field of a Solenoid

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

## Effects of Magnetic Fields

### Force on a Charged Particle

$$F = qvB sin θ$$

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

### Force on a Wire

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

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.

## Physics

Magnetism problem guide

## Circuits

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).

### RL Circuits

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

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.

### RLC Circuits

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.

## Important Equations

$$\epsilon= \frac{\Delta \phi}{\Delta t}$$

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

$$\epsilon = L \frac{\Delta I}{\Delta t}$$

## Example Problems

### Example 1

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

## Example Problems

### Solution

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

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

### Example 2

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?

### Solution

$$\phi = NB . A$$

$$A = Ix$$

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

$$\epsilon = Blv$$