Magnetic Forces on Moving Electric Charges

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A moving charge q of velocity v in a magnetic field B experiences a force F perpendicular to both v and B and whose magnitude is given by: (v, B and F are vectors).

F = q v B sin(θ)

where θ is the angle between v and B.
If v is perpendicular to B, then the above simplifies to

F = q v B

For a positive charge, the direction of the force F is given by the right hand rule as shown below.
- Point the index finger in the direction of motion of the charge
- Point the middle finger in the direction of the magnetic field
and the thumb gives the direction of the force.
For a negative charge, do the same and then the direction of the force will be in the opposite direction to which the thumb is pointing.

right hand rule of force on charge in a magnetic field

Example 1: In a 3-d (x,y,z) system of axes, a proton moving along the positive x - axis at a velocity of 1.3×105 m/s, enters a magnetic field whose magnitude is 0.22 ×10-5 Tesla and field lines are directed along the negative y - axis. What is a) the magnitude and b) direction of the force acting on the moving proton?

Solution:
a) Magnitude
A proton has a charge of 1.6×10-19 C. The angle between the direction of the field and the motion of the proton is 90°. Hence the force F acting on the proton is given by
F = q v B sin(90°)
where q is the charge, v is the velocity of the charge and B is the magnitude of the magnetic field.
F = (1.6×10-19 C) × (1.3×105 m/s) × (0.22×10-5 T) = 0.458 N
b) Direction
The use of the right hand rule yields the result below where the force F is directed downward.

right hand rule for example 1

Example 2: An upward force of 2.5×10-10 N acts on an electron while traveling in a magnetic field of 7.5×10-3 T directed to the East. a) What is the magnitude and b) direction of the velocity of the electron if it is moving perpendicular to the magnetic field?

Solution:
a) An electron has a charge of - 1.6×10-19 C. The angle between the direction of the field (East) and the motion of the electron is 90° (given). Hence the force F due to a magnetic field B an electron moving at a velocity v is given by
F = q v B sin(90°) = q v B
Using the magnitudes of all three vectors v, B and F and the absolute value of q we can write
|F| = |q| |v| |B|
The magnitude of v is given by

|v| =  
|F|
|q| |B|
=  
2.5×10-10 N
( 1.6×10-19 C) ( 7.5×10-3 T)


= 2.1×1011 m/s
b) Direction
First use vectors to draw an upward force F and a magnetic field B to the East (given above).
example 2 given directions
Velocity v is perpendicular to F and also to B (given) and therefore may either be directed out of the page or into the page. Since F is given, use the right hand rule twice for each of the possible directions of v (in and out of the page), and the right answer is the the one that agrees with the right hand rule as on the left of the figure below. On the right, the convention that a vector into the page is represented by a cross.
answer example 2

Example 3: The figure below shows that the velocity v of a moving negative charge is directed into the page. A magnetic field B is directed downward. What is the direction of the force F acting on the charge?

figure for example 3

Solution:

Draw vectors v (into the page) and B and use the right hand rule and take the opposite direction because the charge is negative as shown in the left side of the figure. On the right, the convention that a vector into the page is represented by a cross is used.

solution for example 3

Example 4: What is the direction of the force acting on a negative charge moving downward in a magnetic field directed into the page as shown in the figure below?

figure for example 4

Solution:

Draw vectors v and B (into the page) then use the right hand rule. Because the charge is negative take the opposite direction given by the rule as shown below. On the right, the convention that a vector into the page is represented by a cross is used.

solution for example 4

Example 5: What is the direction of the force acting on a positive charge moving out of the page in a magnetic field directed to the West as shown in the figure below?

figure for example 5

Solution:

Draw vectors v (out of the page as shown in the given figure) and B then use the right hand rule as shown below. On the right, the convention that a vector out of the page is represented by a dot is used.

solution for example 5

Example 6: A force acting on a positive charge is directed out of the page and the magnetic field is directed to the West. Find the direction of motion of the charge if it is moving perpendicular to the magnetic field.

figure for example 6

Solution:

In this example we are given the force F. Draw vectors F (out of the page as shown in the given figure) and B. v can either be up or down since it is perpendicular to both F and B. Since F is given, use the right hand rule twice for each of the possible directions of v (up and down), and the right answer is the the one that agrees with the right hand rule as shown below.

solution for example 6