Magnetic Field due to

a Current-Carrying Wire

Biot-Savart Law

AP Physics C

Mrs. Coyle

Hans Christian Oersted, 1820

• Magnetic fields are caused by currents.• Hans Christian Oersted in 1820’s showed that

a current carrying wire deflects a compass.

No Current in the WireCurrent in the Wire

Right Hand Curl Rule

Magnetic Fields of Long Current-Carrying

Wires

B = o I

2r I = current through the wire (Amps)

r = distance from the wire (m)

o = permeability of free space

= 4 x 10-7 T m / A

B = magnetic field strength (Tesla)

I

Magnetic Field of a Current Carrying Wire

• http://www.walter-fendt.de/ph14e/mfwire.htm

What if the current-carrying wire is not straight? Use the Biot-Savart Law:

20 ˆ

4 r

rdsB

Id

Note: dB is perpendicular to ds and r

Assume a small segment of wire ds causing a field dB:

Biot-Savart Law allows us to calculate the Magnetic Field Vector

• To find the total field, sum up the contributions from all the current elements I ds

• The integral is over the entire current distribution

24

ˆIoμ d

π r

s rB

2

0 ˆ

4i

iiIr

rdsB

Note on Biot-Savart Law • The law is also valid for a current consisting

of charges flowing through space

• ds represents the length of a small segment of space in which the charges flow.

• Example: electron beam in a TV set

Comparison of Magnetic to Electric Field

Magnetic Field

• B proportional to r2

• Vector

• Perpendicular to FB , ds, r

• Magnetic field lines have no beginning and no end; they form continuous circles

• Biot-Savart Law• Ampere’s Law (where

there is symmetry

Electric Field

• E proportional to r2 • Vector

• Same direction as FE

• Electric field lines begin on positive charges and end on negative charges

• Coulomb’s Law• Gauss’s Law (where

there is symmetry)

Derivation of B for a Long, Straight Current-Carrying Wire

Integrating over all the current elements gives

2

1

1 2

4

4

Isin

Icos cos

θo

θ

o

μB θ dθ

πaμ

θ θπa

sin ˆˆd dx θ s r k

If the conductor is an infinitely long, straight wire, = 0 and =

• The field becomes:

2

IoμB

πa a

B for a Curved Wire Segment

• Find the field at point O due to the wire segment A’ACC’:

B=0 due to AA’ and CC’

Due to the circular arc:

• s/R, will be in radians

4

IoμB θ

πR

24

ˆIoμ d

π r

s rB

B at the Center of a Circular Loop of Wire

• Consider the previous result, with = 2

I I

I

24 4

2

o o

o

μ μB θ π

πR πRμ

BR

Note• The overall shape of the magnetic field of the circular

loop is similar to the magnetic field of a bar magnet.

B along the axis of a Circular Current Loop

• Find B at point P

2

32 2 22

Iox

μ RB

x R

24

ˆIoμ d

π r

s rB

If x=0, B same as at center of a loop

If x is at a very large distance away from the loop.

x>>R:

2 2

3 32 2 2 22

I Io ox

μ R μ RB

xx R

Magnetic Force Between Two Parallel Conductors

• The field B2 due to the current in wire 2 exerts a force on wire 1 of

F1 = I1ℓ B2

1 21 2

I IoμF

πa

I 2

2 2oμ

Bπa

Magnetic Field at Center of a SolenoidB = o NI

L

N: Number of turnsL: Length

n=N/L

------------------------L----------------

Direction of Force Between Two Parallel Conductors

If the currents are in the:

–same direction the wires attract each other.

–opposite directions the wires repel each other.

Magnetic Force Between Two Parallel Conductors, FB

• Force per unit length: 1 2

2

I IB oF μ

πa

Definition of the Ampere

• When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

Definition of the Coulomb

• The SI unit of charge, the coulomb, is defined in terms of the ampere

• When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C

Biot-Savart Law: Field produced by current carrying wires

– Distance a from long straight wire

– Centre of a wire loop radius R

– Centre of a tight Wire Coil with N turns

• Force between two wiresa

II

l

F

2

210

a

IB

20

R

IB

20

R

NIB

20

20 ˆ

4 r

rdsB

Id