## Sunday, March 23, 2008

### Concept Review Ch.34 Magnetic Field

Definition of magnetic field

Magnetic field exerts force on a charge particle

Some facts about the magnetic force

a) From a point P, a charge particle can move in any direction or along any line. Along one of these possible lines, if the charge is moving, there is no magnetic force. Magnetic force is defined to be acting along this line.

b) The magnitude of the magnetic force is proportional to the product of speed of the charged particle v and sinθ, θ being the angle the speed makes with the line along which magnetic field is acting. Hence magnetic force is proportional to |v*sinθ|

c) The direction of the magnetic force is perpendicular to the direction of the magnetic field as well as to direction of the velocity.

d) The magnetic force is also proportional to the magnitude of charge q.

e) Its direction is different and opposite for positive and negative charges.

Magnetic field can be defined mathematically as

Vr(F) = qVr(v) × Vr(B) (34.1)

Equation uniquely determines the direction of magnetic field B from the rules of the vector product.

Units of magnetic field

The SI unit of magnetic field is newton/ampere-metre. It is written as Tesla.

Tesla is newton/ampere-metre. Tesla is also defined as weber/m².

Another unit in common use is gauss .

1 T = 104 gauss

We have magnetic field of the order of 10-5 near the earth's surface.

superconducting magnets can create a magnetic field of the order of 10 T.

Earlier, the concept of magnetic field was referred to as magnetic induction.

Electromagnetic field

Electric field and magnetic field are not basically independent. They are two aspects of same entity electromagnetic field. Whether th electromagnetic field will show up as an electric field or a magnetic field or a combination depends on the frame from which we are looking at the field.

Motion of a Charged particle in a uniform magnetic field.

Magentic force on a charged particle is perpendicular to its velocity. Hence there will not any change in its speed or kinetic energy.

The magnetic force will deflect the particle without changing speed and in a uniform field, the particle will move along a circle perpendicular to the magnetic field. The conclusion is that, the magnetic force provides centripetal force.
If r be the radius of the circle,

qvB = mv²/r (LHS is the expression for magnetic force and RHS is expression mass * acceleration)

r = mv/qB ...(34.2)

The time taken to complete the circle is
T = 2πr/v = 2πm/qB ... (34.3)

The time period or time taken to complete one circle is independent of speed.
But the radius (34.2) depends on v. Hence if speed increases, the radius is larger.

Frequency of revolutions is

ν = 1/T = qB/2πm ... (34.4)
This frequency is called cyclotron frequency.

If the velocity of charge is not perpendicular to the magnetic field, the resultant path will be a helix.

The radius of the path will be determined by velocity component which is perpendicular to the magnetic field.

Magnetic Force on a current carrying wire

In a current carrying wire, electrons, which are charge carrying particles are moving and hence in a magnetic field, a current carrying conductor would experince magnetic force.

Vr(F) = iVR(l)×Vr(B) ...(34.6)
Vr is used to denote vector.
The quanity iVr(l) denotes current element of length of l.

Torque on a current loop
If there is a rectangular loop carrying current i in a uniform magnetic field B
then net torque acting on the loop is

Г = iABsinθ
Where i = current in the loop
A = area
B = magnetic field
θ = the angle of inclination of the loop with the plane perpendicular to the plane of magnetic field.

We can also define

Vr(Г) = iVr(A)× Vr(B) ...(34.7)

iVr(A) can be termed as Vr(μ) the magnetic dipole moment of simply magentic moment of the current loop.
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In the material below vectors shown in bold letters.

Definition of magnetic field

Magnetic field exerts force on a moving charged particle

Some facts about the magnetic force

a) From a point P, a charged particle can move in any direction or along any line. Along one of these possible lines, if the charge is moving, there is no magnetic force. Magnetic force is defined to be acting along this line.

b) The magnitude of the magnetic force is proportional to the product of speed of the charged particle v and sinθ, θ being the angle the speed makes with the line along which magnetic field is acting. Hence magnetic force is proportional to |v*sinθ|

c) The direction of the magnetic force is perpendicular to the direction of the magnetic field as well as to direction of the velocity.

d) The magnetic force is also proportional to the magnitude of charge q.

e) Its direction is different and opposite for positive and negative charges.

Magnetic field can be defined mathematically as

F = qv × B

Equation uniquely determines the direction of magnetic field B from the rules of the vector product.

Units of magnetic field

The SI unit of magnetic field is newton/ampere-metre. It is written as Tesla.

Tesla is newton/ampere-metre. Tesla is also defined as weber/m².

Another unit in common use is gauss .

1 T = 104 gauss

We have magnetic field of the order of 10-5 near the earth's surface.

Superconducting magnets can create a magnetic field of the order of 10 T.

Earlier, the concept of magnetic field was referred to as magnetic induction.

Electromagnetic field

Electric field and magnetic field are not basically independent. They are two aspects of same entity electromagnetic field. Whether the electromagnetic field will show up as an electric field or a magnetic field or a combination depends on the frame from which we are looking at the field.

Motion of a Charged particle in a uniform magnetic field.

Magnetic force on a charged particle is perpendicular to its velocity. Hence there will not any change in its speed or kinetic energy.

The magnetic force will deflect the particle without changing speed and in a uniform field, the particle will move along a circle perpendicular to the magnetic field. The conclusion is that, the magnetic force provides centripetal force.
If r be the radius of the circle,

qvB = mv²/r (LHS is the expression for magnetic force and RHS is expression mass * acceleration)

r = mv/qB

The time taken to complete the circle is
T = 2πr/v = 2πm/qB

The time period or time taken to complete one circle is independent of speed.
But the radius depends on v. Hence if speed increases, the radius is larger.

Frequency of revolutions is

ν = 1/T = qB/2πm ... (34.4)
This frequency is called cyclotron frequency.

If the velocity of charge is not perpendicular to the magnetic field, the resultant path will be a helix.

The radius of the path will be determined by velocity component which is perpendicular to the magnetic field.

Magnetic Force on a current carrying wire

In a current carrying wire, electrons, which are charge carrying particles are moving and hence in a magnetic field, a current carrying conductor would experience magnetic force.

If a straight wire of length l carry8ng a current i is placed in a uniform magnetic field B, the force on it is

F = il×B

The quantity il denotes current element of length of l.

Torque on a current loop

If there is a rectangular loop carrying current i in a uniform magnetic field B
then net torque acting on the loop is

Г = iABsin θ

Where i = current in the loop
A = area
B = magnetic field
θ = the angle of inclination of the loop with the plane perpendicular to the plane of magnetic field.

We can also define

Г = iA× B

iA can be termed as μ the magnetic dipole moment or simply magnetic moment of the current loop.

If there are n turns in the loop, each turn experiences a torque.

The net torque is
Г = niA× B

μ = niA