4 magnetic field due to a current-carrying wire, biot-savart law apc

8/12/2019 4 Magnetic Field Due to a Current-Carrying Wire, Biot-Savart Law APC

1/24

Magnetic Field

dueto

a Current-Carrying Wire

Biot-Savart

Law

AP Physics C

Mrs. Coyle

Hans Christian Oersted, 1820


2/24

Magnetic fields are caused by currents.

Hans Christian Oersted in 1820s showed

that a current carrying wire deflects acompass.No Current in the Wire

Current in the Wire


3/24

Right Hand Curl Rule


4/24

Magnetic Fields of Long Current-Carrying

Wires

B = mo I

2p rI = current through the wire (Amps)

r = distance from the wire (m)

mo = permeability of free space

= 4px 10-7T m / A

B = magnetic field strength (Tesla)

I


5/24

Magnetic Field of a Current

Carrying Wire

http://www.walter-

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


6/24

What if the current-carrying wire is not

straight? Use the Biot-Savart Law:

2

0

4 r

rdsB

Id

p

m

Note:

dBis perpendicular to ds and r

Assume a small segment ofwire dscausing a field dB:


7/24

Biot-Savart Law allows us to calculate

the Magnetic Field Vector

To find the total field, sum up the contributionsfrom all the current elements Ids

The integral is over the entire current distribution

24

Io d

r s rB

2

0

4 i

iiI

r

rdsB

p

m


8/24

Note on Biot-Savart Law

The law is also valid for a current consisting

of charges flowing through space

dsrepresents the length of a small segmentof space in which the charges flow.

Example: electron beam in a TV set


9/24

Comparison of Magnetic to Electric Field

Magnetic Field

B proportional to r2

Vector

Perpendicular to FB, ds, r

Magnetic field lines haveno beginning and no end;

they form continuous

circles

Biot-Savart Law

Amperes Law (where

there is symmetry

Electric Field

E proportional to r2

Vector

Same direction as FE

Electric field lines beginon positive charges and

end on negative charges

Coulombs Law

Gausss Law (where

there is symmetry)


10/24

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

o

o

B d

a

a

sin d dx

s r k


11/24

If the conductor is an infinitely long,

straight wire, q1= 0 and q2= p

The field becomes:

2IoB

a a


12/24


13/24

B for a Curved Wire Segment

Find the field at point Odue to the wire segment

AACC:

B=0 due to AA and CC

Due to the circular arc:

qs/R,will be in radians

4

Io

B R

24

Io

d

r

s rB


14/24

B at the Center of a Circular Loop of Wire

Consider the previous

result, with q= 2p

I I

I

24 4

2

o o

o

B

R R

B

R


15/24

Note The overall shape of the magnetic field of

the circular loop is similar to the magneticfield of a bar magnet.


16/24

B along the axis of a Circular Current Loop

Find Bat pointP

2

32 2 2

2

Io

x

RB

x R

24

Io

d

r

s rB

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


17/24

If x is at a very large distance away

from the loop.

x>>R:

2 2

3 3

2 2 2 22

I Io o

x

R RB

xx R


18/24

Magnetic Force Between Two Parallel

Conductors

The field B2due to

the current in wire 2

exerts a force on

wire 1 ofF1= I1B2

1 2

12

I Io

F

a

I 2

22

o

B

a


19/24

Magnetic Field at Center of a SolenoidB = mo NI

L

N: Number of

turnsL: Length

n=N/L

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


20/24

Direction of Force Between Two

Parallel Conductors

If the currents are in the:

samedirection the wires attracteach other.

oppositedirections the wires repeleach other.


21/24

Magnetic Force Between Two

Parallel Conductors, FB

Force per unit length: 1 2

2

I IB o

F

a


22/24


23/24

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


24/24

Biot-Savart Law: Field produced by current

carrying wires

Distance afrom long straight wire

Centre of a wire loop radius R

Centre of a tight Wire Coil with Nturns

Force between two wires

a

II

l

F

p

m

2

210

a

IB

p

m

2

0

R

IB

2

0m

R

NI

B 2

0m

2

0

4 r

rdsB

Id

p

m

4 magnetic field due to a current-carrying wire, biot-savart law apc

Documents