The figure shows three points A, B and C in a uniform electric field \( (\vec{E}) \). The line AB is perpendicular to BC and BC is parallel to \( \vec{E} \). If \( V_A \), \( V_B \) and \( V_C \) are the potentials at A, B and C respectively, then the correct option is
Show Hint
Remember that electric potential decreases in the direction of the electric field. Equipotential lines are perpendicular to the electric field lines. Points on the same equipotential line have the same electric potential. Analyze the orientation of points A, B, and C with respect to the uniform electric field to determine the relationship between their potentials.
In a uniform electric field, the electric potential decreases in the direction of the electric field. The electric field lines point from higher potential to lower potential.
Given that BC is parallel to the electric field \( \vec{E} \), and the electric field lines are directed from left to right, the potential at point B will be higher than the potential at point C. Therefore, \( V_B>V_C \).
The line AB is perpendicular to BC, which means AB is also perpendicular to the electric field \( \vec{E} \). Points that are at the same perpendicular distance from the electric field lines have the same electric potential. Since points A and B lie on the same equipotential line (a line perpendicular to the electric field), their electric potentials are equal. Therefore, \( V_A = V_B \).
Combining these two results, we have \( V_A = V_B \) and \( V_B>V_C \). This implies \( V_A>V_C \).
So, the correct relation between the potentials at points A, B, and C is \( V_A = V_B>V_C \).
Looking at the options, option (B) states \( V_A = V_B>V_C \), and option (D) states \( V_A>V_B = V_C \). There seems to be a slight inconsistency in the provided correct answer and the deduction. Let's re-examine the situation.
Since BC is parallel to \( \vec{E} \) and in the direction of \( \vec{E} \), the potential decreases along BC. Thus, \( V_B>V_C \).
Since AB is perpendicular to \( \vec{E} \), points A and B are at the same potential. Thus, \( V_A = V_B \).
Combining these, we get \( V_A = V_B>V_C \).
The provided correct answer is option (D) \( V_A>V_B = V_C \). This would imply that points B and C are at the same potential, which contradicts the fact that they lie along the direction of the electric field.
Let's assume there might be a misunderstanding in interpreting the diagram or the intended relationship. Given the standard properties of electric fields and potentials:
- Potential decreases along the direction of the electric field.
- Points on a line perpendicular to the electric field are at the same potential.
Based on these, \( V_A = V_B \) because AB is perpendicular to \( \vec{E} \), and \( V_B>V_C \) because B is at a higher potential than C as the electric field points from B towards C. Therefore, \( V_A = V_B>V_C \).
There seems to be an error in the provided correct answer. The logically derived answer is \( V_A = V_B>V_C \).
Final Answer: The final answer is $\boxed{V_A = V_B>V_C}$
