Q.Two identical point charges of magnitude $-q$ are fixed as shown in the figure below. A third charge $+q$ is placed midway between the two charges, at the point P. Suppose this charge $+q$ is displaced a small distance from the point P, in the directions indicated by the arrows $A_1$, $A_2$ (along the line joining the two $-q$ charges) and $B_1$, $B_2$ (perpendicular to that line). In which direction(s) will $+q$ be stable with respect to the displacement?

Step 1. At the midpoint P, the fields of the two equal $-q$ charges cancel exactly, so $+q$ sits in equilibrium there.

Step 2. Displace $+q$ slightly along the line joining the two $-q$ charges (directions $A_1$, $A_2$). Moving toward one $-q$ brings $+q$ closer to that charge and farther from the other; since $+q$ is attracted toward each $-q$, the nearer charge now pulls harder than before and the farther one pulls less, so the net force pulls $+q$ back toward P. This is a restoring force, so the equilibrium is stable along $A_1$, $A_2$.

Step 3. Displace $+q$ slightly perpendicular to the line (directions $B_1$, $B_2$). By symmetry the two attractive forces from the $-q$ charges now both acquire a component pushing $+q$ further away from the line, since each $-q$ pulls $+q$ toward itself and the geometry tips both pulls in the same sideways sense away from P. This is a force that grows the displacement rather than opposing it, so the equilibrium is unstable along $B_1$, $B_2$.

Step 4. A configuration can be stable along one axis and unstable along the perpendicular axis simultaneously (a saddle point), which is exactly the case here.

Displace the test charge along and perpendicular to the line joining the fixed charges and check whether the resulting net force restores or grows the displacement.

• Assuming a point of zero net force is automatically stable in every direction, without checking each direction separately.
• Forgetting that for two like charges flanking a positive test charge, the equilibrium is a saddle point (stable one way, unstable the other), not a simple stable or unstable point.