Example: What is the magnetic field at the center of a circular arc of wire?

02

ˆ

4

Ids rdB

r

s R ds Rd

ˆ 1 sin 90ds r ds 1

02ˆ

4

I dsdB k

r

Ids

R

cx

y

02ˆ

4

I RddB k

R

0 ˆ4

Id k

R

0

0

ˆ4

IB d k

R

0

0

ˆ4

IB d k

R

0 ˆ4

IB k

R

This will work for any current carrying arc, where q (in radians) is the angle that defines the arc you are looking at. The direction would be perpendicular to the plane containing the arc.

This is the most common use for the Biot – Savart Law, since the resulting expression is relatively easy to use.

I

We have seen that the direction of the magnetic field from a current carrying wire is perpendicular to the direction of the current and to the position vector to a specified point. What then would the magnetic field lines look like?

I

x x x x xI

B

We use the second right-hand rule to determine the direction of the magnetic field.1. Point thumb in direction of current.2. Fingers show direction of the magnetic field.

What would the magnetic field lines look like for a single loop of wire?

Circles

B

I

x

x

x xx

B

x

xx

A single current carrying wire will create a magnetic field. What happens when a second current carrying wire is brought near the first wire?

There would be an interaction force between the two magnetic fields.

What would the magnitude of this force be?

12 2 2 1F I L B

2 2 1 sinI L B 1

0 12 2 2

II L

r

0 112 2 2 2

IF I L

r

Current through wire 2 and length segment of wire 2

Magnetic field due to wire 1 Current through wire 1

Distance from wire 1 to the location of wire 2

Force on one current carrying wire due to the other.

What about direction?

I

B

I

BI

B

xI

B

Parallel Wires Anti-parallel Wires

N S

NS

N

S

N

S

FF FF

- Attract - Repel

On a computer chip, two conducting strips carry charge from P to Q and from R to S. If the current direction is reversed in both wires, the net magnetic force of strip 1 on strip 2

1. remains the same.2. reverses.3. changes in magnitude, but not in direction.4. changes to some other direction.5. other

We are now going to discuss the second method for determining the strength of a magnetic field from a current carrying wire.

0 encircledB ds I

Ampere’s Law

Iencircled – amount of current encircled by the Amperian loop.B – Magnetic field strength at the location of the Amperian loop.ds – segment of the line defining the Amperian loop.

This method is used for systems that are highly symmetric, because it is much easier to use than the Biot – Savart Law for these cases.

Ampere’s Law is very similar to Gauss’s Law.

Amperian Loop

The current flows through the entire cross-sectional area of the wire, but we only consider the amount of current that is encircled by the Amperian loop.

The amperian loop is an imaginary loop draw at the location that you are interested in determining the magnetic field.

Wire

Magnetic field at Amperian loop.