Magnetic Flux

AP Physics C

Montwood High School

R. Casao

• Magnetic flux is the number of magnetic field lines passing through a plane area A.

• Consider an element of area dA on an surface. If the magnetic field at this element is B, then the magnetic flux through the area is B•dA, where dA is a vector perpendicular to the surface.

• The total magnetic flux m through the surface is:

• Consider a plane of area A and a uniform magnetic field B, which makes an angle with the vector dA:

AdBΦm

θcosABAdBΦm

– If the magnetic field B lies parallel to the plane of the surface, the angle between B and dA is 90°.

– The magnetic flux is zero because no magnetic field lines pass through the plane of the surface.

0Φ

09cosABΦ

θcosABΦ

m

m

m

– If the magnetic field B lies perpendicular to the plane of the surface, the angle between B and dA is 0°.

– The magnetic flux is maximum and the greatest number of magnetic field lines pass through the plane of the surface.

ABΦ

0cosABΦ

θcosAdBΦ

m

m

m

• Remember that the angle is the angle between the magnetic field vector B and the area vector dA.

• The unit for magnetic flux is the weber (Wb).

1 Wb = 1 T·m2

Flux Through a Rectangular Loop

• A rectangular loop of width a and length b is located a distance c from a long wire carrying current I.

• The wire is parallel to the long side of the loop.

• From Ampere’s law, the magnetic field due to the wire at a distance r from the wire is:

• The magnetic field B varies over the loop and is directed into the page.

• B is parallel to dA, so the magnetic flux through an element of area dA is:

rπ2

IμB o

Adrπ2

IμΦ o

m

• The magnetic field is not uniform throughout the area of the rectangular loop; it decreases in magnitude from length c to length c + a.

• Divide the length a into small elements of length dr.

• The area dA = b·dr.

• The contribution of each element of area dA to the total magnetic flux is:

• The total magnetic flux through the rectangular loop is determined by integrating from c to c + a.

drr

1

π2

bIμΦ

drbrπ2

IμΦ

ac

c

o

m

ac

c

o

m

drbrπ2Iμ

dΦ om

• Evaluating the integral:

c

acln

π2

bIμΦ

clnaclnπ2

bIμΦ

rlnπ2

bIμΦ

drr

1

π2

bIμΦ

o

m

o

m

ac

c

o

m

ac

c

o

m

Gauss’ Law in Magnetism• Magnetic field lines are continuous and form

closed loops.

• Magnetic field lines due to currents do not begin or end at any point.

• The magnetic field lines of a bar magnet illustrate the point.– For any closed surface, the number of magnetic field

lines entering the surface is equal to the number leaving the surface.

– The net magnetic flux is 0 T·m2.

• Gauss’s law in magnetism states that the net magnetic flux through any closed surface is always zero.

• Gauss’ law in magnetism is based on the experimental fact that isolated magnetic poles (or monopoles) have not been detected, and may not even exist.

0AdB

• The known sources of magnetic fields are magnetic dipoles (current loops).

• All magnetic field effects in matter can be explained in terms of magnetic dipole moments (effective current loops) associated with electrons and nuclei.

• Magnetic flux for multiple loops:

AdBNΦm