D3-1. Magnetic Fields and Force on a Conductor

1. Right-Hand Grip Rules

Straight Current-Carrying Wire

Magnetic field lines form concentric circles centered on the wire.
The lines are closer together near the wire (stronger field) and further apart away from it (weaker field).
Right-Hand Grip Rule: Point your right thumb in the direction of the current. Your curled fingers show the direction of the magnetic field ($\vec{B}$).

Flat Circular Coils & Solenoids

The magnetic field of a coil or solenoid is similar to a bar magnet.
The field lines are straight through the center and curve around the outside.
Alternate Grip Rule: Curl your right fingers along the direction of the current loop. Your thumb will point towards the North pole of the field.

2. Magnetic Field Properties & The Lorentz Force

The Formula

$$F = qvB\sin\theta \quad \Rightarrow \quad \vec{F} = q(\vec{v} \times \vec{B})$$ $$F = BIL\sin\theta \quad \Rightarrow \quad \vec{F} = L(\vec{I} \times \vec{B})$$

Calculates the magnetic force on a moving charge or current-carrying wire.
Direction: The magnetic force is always perpendicular ($\perp$) to both the velocity/current and the magnetic field.
Maximum/Minimum: Force is maximum when $\theta = 90^\circ$ and zero when parallel ($\theta = 0^\circ$).

Notation Key & Units:

$F$ = Magnetic force ($\text{N}$)
$q$ = Charge ($\text{C}$)
$I$ = Current ($\text{A}$)
$v$ = Velocity ($\text{m s}^{-1}$)
$L$ = Length of wire ($\text{m}$)
$B$ = Magnetic field strength ($\text{Tesla, T}$)
$\theta$ = Angle between $\vec{v}$ (or $\vec{L}$) and $\vec{B}$

Field Properties in 3D:

Field lines always point from the north pole (N) to the south pole (S).
Dots: represent the magnetic field directed out of the plane of the page.
Crosses: represent the magnetic field directed into the plane of the page.

3. Examples

Example 1

Problem: [M17, P1, Q21] A positively-charged particle moves parallel to a wire that carries a current upwards.

[Create image here: Positively charged particle parallel to upward current wire]

What is the direction of the magnetic force on the particle?
A. To the left
B. To the right
C. Into the page
D. Out of the page

Solution:

By the Right-Hand Grip rule, the wire creates a magnetic field that is pointing into the page on the right side where the particle is. Applying the Right-Hand Rule (Cross Product) for a positive charge moving upwards in a field pointing into the page, the resulting force is to the left (towards the wire). The answer is A.

Example 2

Problem: [M16, P1, Q21] A wire carrying a current $I$ is placed in a region of uniform magnetic field $B$, as shown in the diagram.

[Create image here: Wire placed at angle in B field out of page]

The direction of the field $B$ is out of the page, and the length of the wire is $L$. What is correct about the direction and magnitude of the force acting on the wire?

	Direction	Magnitude
A.	$\searrow$	equal to $BIL$
B.	$\searrow$	smaller than $BIL$
C.	$\nearrow$	equal to $BIL$
D.	$\nearrow$	smaller than $BIL$