Gauss's Law for Magnetic Fields

Question 1

How do you determine the direction of a magnetic field due to a current in a wire?

Answer 1

• using the right hand rule
• thumb point in the direction of conventional current
• fingers curl in the direction of the magnetic field

Question 2

Describe the Magnetic Field in a Solenoid

Answer 2

• magnetic field lines are always in a loop
• a solenoid appears to have a north pole at one end and a south pole at the other
• but actually the loops are continuous throughout

Question 3

Solenoids and Divergence

Answer 3

• the divergence of the field of a solenoid is zero everywhere
• in any small volume dV, any field line going in is also going out as the lines are in loops so divergence is zero

Question 4

Magnetic Dipole

Answer 4

• very similar field pattern to an electric dipole
• but the field lines are in loops whereas electric field lines stop and start on charges
• at the centre the magnetic field lines are in the opposite direction to the electric field lines

Question 5

Gauss’s Law for Magnetic Fields - Differential Form

Equation

Answer 5

∇ . |B = 0

Question 6

Gauss’s Law for Magnetic Fields - Integral Form

Description

Answer 6

• the total magnetic flux passing through any closed surface is zero, i.e for every field line entering the volume enclosed by the surface there is also a field line leaving it
• there are no point in space from which a magnetic field line diverges from or converges to
• magnetic monopoles do not exist

Question 7

Gauss’s Law for Magnetic Fields

Derivation

Answer 7

∇ . |B = ΔV->0 lim 1/ΔV ∮|B.n^dA = ϕB
-net flux, ϕB, is telling you whether inside the closed surface you have a net source or a net sink of magnetic field lines and whether the net flow is in or out
-since there are no monopoles the net flux across any closed surface is always zero so divergence must also be zero:
∮|B.n^dA = 0

Question 8

What is a vector field with zero divergence called?

Answer 8

• solenoidal fields
• all magnetic fields are solenoidal
• any vector field can be written as the sum of an irrotational field and a solenoidal field

Question 9

Biot-Savart Law

Magnetic Field of a Moving Point Charge

Answer 9

|B(|r) = μ0/4π * q|v x |r /r²

Question 10

Biot-Savart Law

Magnetic Field of a Steady Current

Answer 10

|B(|r) = I*μ0/4π * ∫ {d|I’ x |r / r²}

|I' = an element of length along the wire
|r = the vector from the source to the point P

Question 11

Biot-Savart Law

Magnetic Field of an Infinite Wire

Answer 11

B = I*μ0/2πa

a = perpendicular distance from the wire

Question 12

Lorentz Force

Magnetic Field

Answer 12

|Fmag = Q(|v x |B)

The force Fmag, on a charge Q moving with speed v through a magnetic field B

Question 13

Lorentz Force

Magnetic and Electric Field

Answer 13

|F = Q{|E + (|v x |B)}

The force |F, on a charge Q moving with speed v through an electric field E and a magnetic field B

Question 14

Gauss’s Law for Magnetic Field’s - Integral Form

Equation

Answer 14

∮|B.n^dA = 0

Question 15

Difference Between Electric and Magnetic Fields

Answer 15

• direction of magnetic field is perpendicular to the magnetic force, direction of electric field is parallel or antiparallel to force
• for determining magnetic field, speed and direction of the test charge must be considered
• for magnetic field, the component of the magnetic force in the direction of displacement is always zero
• electrostatic fields are produced by electric charges whereas magnetostatic fields are produced by electric currents

Question 16

Similarities Between Electric and Magnetic Fields

Answer 16

• field is proportional to fields

- the field is defined by the force on a test charge

Question 17

Sketching Magnetic Fields

Answer 17

• magnetic fields don’t originate and terminate on charges, they form closed loops
• field lines appear to originate on the north pole of a magnet and terminate on the south pole of a magnet, the field lines actually pass all the way through the magnet in closed loops
• the net magnetic field is the vector sum of all magnetic fields at that point
• magnetic field lines never cross

Question 18

Magnetic Field of a Solenoid

Equation

Answer 18

|B = μoNI/L ^x

Question 19

Magnetic Field of a Torus

Equation

Answer 19

|B = μoNI/2πr ^φ

Question 20

Magnetic Flux

uniform B perpendicular to S

Answer 20

ΦB = | |B | * surface area

Question 21

Magnetic Flux

uniform B at an angle to S

Answer 21

ΦB = |B . |n * surface area

Question 22

Magnetic Flux
(non-uniform B at variable angle to S)

Answer 22

ΦB = ∫ |B.n^dA

Question 23

Gauss’s Law for Magnetic Fields - Differential Form

Description

Answer 23

• the divergence of a magnetic field is zero everywhere
• since there are no point sources or sinks of magnetic field, the amount of incoming field must equal the amount of outgoing field everywhere