[latexpage]

## Notations:

- Out of the screen/paper:
- Into the screen/paper:

## Magnetic field created by a current-carrying wire

Consider the following current-carrying wire:

- General equation describing this system:

- Variables:
- magnetic field created by moving charge
- magnitude of the magnetic field at a point in space outside the wire (SI unit: )

- permeability the material of the wire is made of (SI unit: )
- electric current (SI unit: )
- distance from the current-carrying wire (SI unit: )

- magnetic field created by moving charge

## Magnetic Force: two currents in the same direction

Consider two identical current-carrying wires with charge flowing through them in the same direction:

- Variables
- = distance between the two wires
- and denote the magnetic field and length of wire (respectively)
- Importantly, for the purposes of this article, the direction is defined as the direction current is traveling
- Note: we treat as a scalar value

- Let denote the magnetic force of on
- Then,
- Using right-hand rule convention, this implies that the direction of is pointed purely in the horizontal direction towards the right

- Similarly, if we let deonote the magnetic force of on then…
- Using the right-hand rule convention, this implies that the direction of is pointed purely in the horizontal direction towards the left 3

- Summary of this example:
- is exerting a downward force on
- is exerting an upward force on
- and directions indicate that both wires will become attracted towards eachother
- This will cause (i.e., the distance between the wires) to decrease
- As decreases , the wires wil accelerate towards eachother

## Induced current in a wire

Consider the following magnetic field (purple) popping out of the page and a wire laid over the field so that it is overlapping for a distance

- Let (orange) be a charge positioned at the lower end of the wire (SI units: Coulombs)

Nothing happens when is stationary

- This is because the force due to the magnetic field is equal to the cross product of times the velocity of the charge and :

- Recall: the magnitude of the cross product can be written as…

In our diagram, the wire and the magnetic field are perpendicular to eachother

- Recall:
- Thus, for this instance,

- For our stationary charge,
- Thus,

The work done by a magnetic field per charge in a coductive wire can be described by the following equation:

- Importantly, the output for this equation has units in Joules/Coulomb
- Recall:
- **Thus, the work done by the magnetic field on a charge causes the charge to move AS IF there is a potential differe4nce of over the length of the wire that overlaps with
- In this context, we aren’t saying the charge is moving due to a difference in potential energy (like it does in a simple DC circuit)
- Even though differences in potential energy give rise to voltage, in teh context of our example, it is the magnetic field giving rise to the voltage
- Specifically, when a voltage is induced from a magnetic field, it is said that the magnetic field is exerting an
**electromotive force**or**emf**

- Specifically, when a voltage is induced from a magnetic field, it is said that the magnetic field is exerting an

- emf has units in J/C
- For circuit analysis, this has the same effect as a potential (or voltage) difference
- We can define the emf mathematically as
- Recall: we can pull constants out of a cross product to rewrite this equation as