Q59P - Introduction to Electrodynamics

Question

Prove or disprove (with a counterexample) the following

 

Theorem: Suppose a conductor carrying a net charge Q, when placed in an

external electric field ${\overrightarrow{\mathbf{E}}}_{\mathbf{e}}$ , experiences a force $\overrightarrow{\mathbf{F}}$ ; if the external field is now

reversed ( ${\overrightarrow{\mathbf{E}}}_{\mathbf{e}}\mathbf{\to }\mathbf{-}{\overrightarrow{\mathbf{E}}}_{\mathbf{e}}$ ), the force also reverses ( $\overrightarrow{\mathbf{F}}\mathbf{\to }\mathbf{-}\overrightarrow{\mathbf{F}}$ ).

 

What if we stipulate that the external field is uniform?

Step-by-Step Solution

Answer

If the external electric field near a conductor is reversed, the force on the conductor doesn't always get reversed.

Step 1:Given data:

A conductor carrying a net charge Q is placed in an external electric field  ${\overrightarrow{E}}_e$.

Step 2: Charge induction on a conductor

A charge induces opposite charge on a conductor near it.

Step 3: A spherical conductor placed in front of a point charge

Consider a conducting sphere placed in front of a positive point charge. The point charge induces negative charge on the closest surface of the sphere. The force on the sphere is thus towards the point charge.

 

If the point charge is negative, the field on the sphere changes sign. This time, positive charge is induced on the closest surface. The force is thus still towards the point charge.

 

Thus, in this case, the change in direction of the external electric field doesn't change the direction of force on the conductor. This disproves the theorem.