• READ THE BOOK!!
• Study examples given and read the summaries
• Look at technique sections
• Practice, Practice, Practice...

You must know how to:

• Use Coulomb's law for a charge distribution
• Find the electric field using Coulomb's Law
• Describe how a charged particle moves in an E field
• Find the electric field using Gauss's Law
• Treat special case of a conductor - no E inside
• Find potential energy, U, and electric potential, V
• Derive E from V and vice-versa

READ THE QUESTION CAREFULLY!

1) E due to charged particle distribution

E =  1 4pe0 å i   Qi ri2   ^ ri

Use vector sum for each charge, where for each charge:

• Magnitude of E µ | Q |
• E µ r^-2
• Vector E points away from a positive charge

Example:
Charges +Q, +2Q, -5Q and +2Q are placed at the four corners ABCD respectively of a square of side a. Find the electric field strength at the center of the square.

2) Coulomb's Law for a continuous charge distribution

F12 =  1 4pe0  Q1 Q2 r2   ^ r     ;       E =  F q0 =  1 4pe0  Q r2   ^ r

Continuous may mean a line, ring, sheet or volume.

• Linear charge density, l - units C m^-1
• Surface charge density, s - units C m^-2
• Volume charge density, r - units C m^-3

Follow tips on pp. 712-713

• Draw charge distribution; always include point at
  which E required
• Use Coulomb's Law for each charge or charge segment
• Add contributions using vector sum or integration

Example: (similar to example on p.689)
Show that E_z at distance z along the z-axis from a long, straight, uniform line charge of length 2l centered at the origin and oriented along the z-axis is given by

ll [2pe0 (z2 - l2)]

where z > l.

3) Using Gauss's Law to find E

Fnet = ó (ç) õ S  En dA =  Qinside e0

Most useful if charge distribution is highly symmetric

Follow tips on pp. 712-713

• Select Gaussian Surface carefully (e.g. sphere, cylinder)
• Use appropriate charge distribution (l, s, r)
• Insert into above equation and solve

Study Chapter 23 examples - classic exam-type questions.

Example
Determine E inside and outside a thin, uniformly charged sphere of radius R and total charge Q

Example
Determine E inside and outside a uniform, spherical distribution of charge of radius R and total charge Q

Example:

Consider two concentric shells of radii a and b with uniform surface charge densities s_a and s_b. They have equal and opposite total charge Q_a = -Q and Q_b = +Q where Q > 0.

(a) develop expressions for the electric field in all three regions of space: r < a, a < r < b and r > b, where r is the distance from the centre of the spheres.

(b) draw a graph of E versus r from r=0 to r=3a for the case where b=2a.

Example:

A straight, thin wire filament 12m long with Q = -74 nC and uniform linear charge density is coaxial with a neutral conducting pipe of the same length; the inner radius is 6.0 mm and the outer radius is 9.0 mm.

(a) Estimate the induced surface charge densities of the inner and outer surfaces of the pipe.

(b) For points in the perpendicular bisector plane, make a graph of E versus R in the range 1 to 15 mm, where R is the perpendicular distance from the filament.

V =  U Q0   ;       U(r) = - ó õ r ¥  F ·dl  ;       V(r) = - ó õ r ¥  E ·dl

For a system of charged particles

V =  1 4pe0 å i   qi ri

For a continuous distribution of charge

V =  1 4pe0 ó õ  dq r

We will only deal with simple cases (planes, spheres, etc.)

Need to understand V, U and how to derive E from V.

Example:

Determine the electric potential inside and outside a
uniform, spherical distribution of charge of radius r₀ and
total charge Q

Example

Consider a uniformly charged rod of length 2l and linear charge density l (=Q/2l) centered at the origin and orientated along the z-axis. Show that the potential at points in the perpendicular bisector plane (the xy plane) is

V =  l 2pe0   ln l + Ö l2 + R2 R

where R = Ö{x² + y²}.