Question

An electric field \( E = 3x^2 \, i \, \text{N C}^{-1} \) exists in a certain region of space. The potential difference between the origin and at \( x = 4 \, m \), \( V_0 - V_4 \) is

• A. $-20$ V
• B. $-40$ V
• C. $-64$ V
• D. $64$ V

Hint:

Potential difference can be found using integration of electric field: \( V = \int E \, dx \) (sign depends on limits).

Answer 1

The Correct Option is D

Step 1: Relation between electric field and potential.
\[ E = -\frac{dV}{dx} \]

Step 2: Express potential difference.
\[ V_0 - V_4 = \int_{0}^{4} E \, dx \]

Step 3: Substitute given electric field.
\[ V_0 - V_4 = \int_{0}^{4} 3x^2 \, dx \]

Step 4: Perform integration.
\[ \int 3x^2 dx = x^3 \]

Step 5: Apply limits.
\[ V_0 - V_4 = \left[ x^3 \right]_{0}^{4} = 4^3 - 0 = 64 \]

Step 6: Final conclusion.
\[ \boxed{64 \, \text{V}} \] Hence, correct answer is option (D).