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