Sources of magnetic fields
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Sources of Magnetic Fields. Besides magnetic poles, electric currents create magnetic fields. There are two ways of calculating B produced by currents: Biot-Savart Law : Field of a “current element” (analogous to a point charge in electrostatics).

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Sources of Magnetic Fields

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Sources of magnetic fields

Sources of Magnetic Fields


Sources of magnetic fields

Besides magnetic poles, electric currents create magnetic fields.

  • There are two ways of calculating B produced by currents:

  • Biot-Savart Law: Field of a “current element”

  • (analogous to a point charge in electrostatics).

  • Ampère’s Law: An integral theorem similar to

  • Gauss’s law.


Sources of magnetic fields

Biot-Savart Law

r

dB

r

θ

I

I

ds

(“permeability of vacuum”)

We getthe total B by integrating ds along the wire:


Sources of magnetic fields

Question:

When is dB=0 ?

How does dB vary with r ?

What is the direction of B at points a and b ?

a

Current I in a straight wire

b


Sources of magnetic fields

Example 1

Find B at a distance R from a long straight wire

y

dB

r

f

R

θ

I

x

dx

(note sin θ = R/r = cos f)


Sources of magnetic fields

Solution


Sources of magnetic fields

Result: the magnitude of the field produced at a distance R from a long straight wire is:

The field lines form circles around the wire. Note the right-hand rule.

I

I


Sources of magnetic fields

Quiz

P

a

I

What is the field at point P?

  • m0I/(4pa)

  • m0I/(2pa)

  • m0I/(pa)

  • zero

  • none of the above


Sources of magnetic fields

Example 2

  • Circular loop in y-z plane. Find B.

  • at the origin

  • at point (x, 0, 0)

y

I

R

x

(x, 0, 0)

z


Sources of magnetic fields

Solution

y

At the Origin:

dS

R

r

z

dB

I


Sources of magnetic fields

At the point (x,0,0):

Pick a short segment ds where the wire crosses the y – axis:

I ds

dB

θ

r

R

θ

x

θ

r’

θ

dB’

x

I ds’


Sources of magnetic fields

Solution


Sources of magnetic fields

Field patterns

I

B

B

N

S

current loop

bar magnet


Sources of magnetic fields

Example 3

I

I

θo

R2

R1

O

  • Find the magnetic field at point O due to the current in the circuit. (Use the Biot-Savart Law on each segment)

  • b) Evaluate the expression for:

  • I = 20A, R1 = 10cm, R2 = 30cm, θo=120º


Sources of magnetic fields

Solution


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