Electric field due to charge distributions cont

1
Electric field due to charge distributions contd
2
Class Object

• Reinforce the ideas and problem solving
  strategies used for the example of a rod with a
  uniform charge distribution.

3

• Be able to identify the similarities between the
  strategy for calculating the electric field due
  to a rod and a ring.
• Be able to apply the strategies learnt to
  non-uniform charge distributions and to other
  shapes and configurations.

4
Electric Field due to a ring of uniform charge
5
Electric Field due to a ring of uniform charge

• Consider a ring of radius R, with uniform charge
  Q. What is the total electric field ata point P a
  distance z along an axis through its centre?

6
Electric Field due to a ring of uniform charge

• Consider a ring of radius R, with uniform charge
  Q. What is the total electric field ata point P a
  distance z along an axis through its centre?

ds
7
Electric Field due to a ring of uniform charge

• Applying the concepts from the previous example,
  we consider the charge per unit length.

8
Electric Field due to a ring of uniform charge

• Applying the concepts from the previous example,
  we consider the charge per unit length.
• In this case, .

9
Electric Field due to a ring of uniform charge

• Applying the concepts from the previous example,
  we consider the charge per unit length.
• In this case, .
• The electric field due to a charge element ds is

10
Electric Field due to a ring of uniform charge

• Because of symmetry we that again the horizontal
  components cancel. Therefore we write that,

11
Electric Field due to a ring of uniform charge

• Because of symmetry we that again the horizontal
  components cancel. Therefore we write that,
• Where r and ? are constant.

12
Electric Field due to a ring of uniform charge

• Therefore the integral becomes,

13
Electric Field due to a ring of uniform charge

• Therefore the integral becomes,
• This integral is simply over the circumference so
  that,

14
Electric Field due to a ring of uniform charge

• From the diagram we can substitute for the cosine.

15
Electric Field due to a ring of uniform charge

• From the diagram we can substitute for the
  cosine.

16
Electric Field due to a ring of uniform charge

• After some algebra we write that,