Question

The force on a charged particle moving in a magnetic field can be computed as the vector sum of the forces due to each separate component of the magnetic field. As an example, a particle with charge q is moving with speed v in the? y-direction. It is moving in a uniform magnetic field $\underset{B}{\rightarrow}=B_x i+ B_yj^+ B_zk$

Part A

What is the x-component of the force F?  exerted on the particle by the magnetic field?

Part B

What is the y-component of the force F?  exerted on the particle by the magnetic field?

Part C

What is the z-component of the force F?  exerted on the particle by the magnetic field?

Part D

If q<0 and Bx=By=Bz>0, find the magnitude of F?  in terms of q, v, and Bx.

Part E

If q<0 and B_xvq=B_yvq=B_zvq=1, find the direction of F

Express your answer in terms of unit vectors i, j, and k.

Answer 1

Concepts and reason

The concepts required to solve the problem are the magnetic force due to particle moving in magnetic field, the vector product and the direction and magnitude of the vectors.

Initially, use the charge, velocity vector and the magnetic field to calculate the magnetic force. Later on, choose the components of the field in each direction and finally use the components to calculate the magnitude and direction of the force.

Fundamentals

The magnetic force acting on a charge moving in a magnetic field is,

$F=q(vxB) $

Here, is the force, is the charge of the particle, is the speed, and is the magnetic field.

The force, velocity and the magnetic field are vectors. The vectors have both direction and magnitude.

The force can be represented in vector form as,

Here, is the x component of the force, is the y component of the force, is the z component of the force. , and are unit vectors along x, y and z axis.

The velocity can be represented in vector form as,

Here, is the x component of the velocity, is the y component of the velocity, is the z component of the velocity. , and are unit vectors along x, y and z axis.

The magnetic field can be represented in vector form as,

Here, is the x component of the magnetic field, is the y component of the magnetic field, is the z component of the magnetic field. , and are unit vectors along x, y and z axis.

The cross product of the velocity vector and the magnetic field is,

The cross product of magnetic force is,

The x component of the magnetic force is,

The y component of the magnetic force is,

The z component of the magnetic force is,

The magnitude of the force is,

(A)

The x component of the magnetic force acting on the particle is,

Substitute for and for . The x component of the magnetic force is,

The x component of the magnetic force acting on the particle is .

(B)

The y component of the magnetic force acting on the particle is,

Substitute for and for . The y component of the magnetic force is,

The y component of the magnetic force acting on the particle is .

(C)

The z component of the magnetic force acting on the particle is,

Substitute for and for . The z component of the magnetic force is,

The z component of the magnetic force acting on the particle is .

(D)

The magnitude of the force is,

Substitute for , for and for . The magnitude of magnetic force is,

For a charge,

Substitute for in to find the magnitude of magnetic force.

The magnitude of magnetic force is,

(E)

The force can be represented in vector form as,

Substitute for , for and for . The magnetic force is,

For , .

Substitute 1 for and 1 for in to find the direction of magnetic force.

Ans: Part A

The x component of the magnetic force acting on the particle by the magnetic field is .

Part B

The y component of the magnetic force acting on the particle by the magnetic field is .

Part C

The z component of the magnetic force acting on the particle by the magnetic field is.

Part D

The magnitude of the magnetic force is .

Part E

The direction of magnetic force is .