Question

Two charges q1 and q2 are placed at points A and B respectively at a distance r away from each other in an external electric field E. The potential energy of the system in the electric field is U? Assume potential energy to be zero when they were a large distance apart and not in the range of electric field E. Now the position of the two charges is interchanged. What is the potential energy of the system in this electric field E?
1. Greater than U
2. Equal to U
3. Less than U
4. Cannot be determined with the given information

Answer 1

Correct Answer - Option 4 : Cannot be determined with the given information

CONCEPT:

•  Force one a charge q in an electric field is defined as 

⇒ F = Eq

• The work done by this force in moving the charge by a distance d along the direction of electrostatic force (along the electric field) is

⇒ W = F.d = Eq.d

• The work done by external force in moving this charge by a distance d along the direction of the electrostatic field is

⇒ Wext = - W = - Eq.d

• By definition, the change in potential energy in moving the charge in an electrostatic field is equal to the work done by external forces

\(\Rightarrow \bigtriangleup U = W_{ext} = -Eq.d\)

• Electric potential is equal to the amount of work done per unit charge by an external force to move the charge q from infinity to a specific point in an electric field

\(\Rightarrow V=\frac{W_{ext}}{q}\)

• Therefore the relation between electric potential and electric potential energy is given by

\(\Rightarrow \bigtriangleup U = W_{ext} = Vq\)

EXPLANATION:

• The potential energy of the system is due to the interaction between q₁ and q₂ 

\(\Rightarrow U = W_{ext} = Vq\)

Where V is the potential at second charge due to the first charge and q is the second charge

\(\Rightarrow U_1 = Vq = \frac {q_1 }{4 \pi \epsilon_0r}q_2\)

• The potential energy of the system is due to the interaction between q₁ and E

\(\Rightarrow U_2 = W_{ext} = Vq = (V_A)q_1\)

Where V_A is is the potential at A due to electric field E

• The potential energy of the system is due to the interaction between q₂ and E 

\(\Rightarrow U_3 = W_{ext} = Vq = (V_B)q_2\)

Where V_B is is the potential at B due to electric field E

• ​It can be observed that U₁, U₂, and U₃ are functions of q₁, q₂, V_A, and V_B. Therefore the change in potential energy cannot be determined unless these functions are known. Therefore option 4 is correct.