Assume that an electric field \( (20x^{4}\,\hat{\imath})\,\mathrm{V\,m^{-1}} \) exists in the region of space. If \( V_0 \) is the potential at the origin and \( V_A \) is the potential at \( x = 1\,\mathrm{m} \), then the potential difference \( V_A - V_0 \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( -4\,\mathrm{V} \)
The potential difference between two points is the negative of the line integral of the electric field. The field here varies with \( x \), so it must be integrated, not multiplied by the distance.
\( \displaystyle V_A - V_0 = -\int_{0}^{1} E\,dx = -\int_{0}^{1} 20x^{4}\,dx \)
\( \displaystyle V_A - V_0 = -\left[4x^{5}\right]_{0}^{1} = -(4 - 0) = -4\,\mathrm{V} \)
The sign is negative because the field points along \( +x \) and the potential falls in the direction of the field.