54 Week 1: Discrete Charge and the Electrostatic Field
Problem 3.
y−a+aqqxq 0Two equal positive charges are on theyaxis, one aty= +aand the other aty=−a. The electric
field at the origin is zero. A test chargeq 0 placed at the origin will therefore be in equilibrium.
a) Discuss the stability of the equilibrium for a positive test charge byconsidering small displace-
ments from equilibrium along thexaxis and small displacements along theyaxis.b) Repeat part (a) for a negative test charge.c) Find the magnitude and sign of a chargeq 0 that when placed at the origin results in a net
force of zero on each of the three charges. What will happen if anyof the charges are dis-
placed slightly from equilibrium in different directions (is the equilibrium stable, unstable,
metastable)?The point at the origin is called asaddle pointbecause the potential there is shaped like asaddle,
with a smooth minimum along one axis and a smooth maximum along the axisperpendicular to it.
Bear this in mind for a couple of weeks until we define and evaluate electrostatic potential!