Question

Electric field in a certain region is given by \( \vec{E} = \left( \frac{A}{x^2} \hat{i} + \frac{B}{y^3} \hat{j} \right) \). The SI unit of A and B are:

• A. Nm\(^3\)C\(^{-1}\); Nm\(^2\)C\(^{-1}\)
• B. Nm\(^2\)C\(^{-1}\); Nm\(^3\)C\(^{-1}\)
• C. Nm\(^3\)C; Nm\(^2\)C
• D. Nm\(^2\)C; Nm\(^3\)C

Hint:

In electromagnetism, the electric field is expressed in terms of its components in different directions. The units of the constants in these equations depend on the power of the distance variables in the denominator and the overall unit of electric field.

Answer 1

The Correct Option is B

Step 1: The electric field is given as: \[ \vec{E} = \left( \frac{A}{x^2} \hat{i} + \frac{B}{y^3} \hat{j} \right) \] Here, \( \vec{E} \) has units of electric field, which in SI is measured in volts per meter (V/m), or equivalently, Newtons per Coulomb (N/C). 
Step 2: Determining the units of A and B. We know that electric field \( \vec{E} \) has units of N/C. Let’s consider each component: 1. For the \( \hat{i} \) component: \[ \frac{A}{x^2} \quad {has units of} \quad \frac{A}{{m}^2} \quad \Rightarrow \quad A = {Nm}^2{C}^{-1} \] 2. For the \( \hat{j} \) component: \[ \frac{B}{y^3} \quad {has units of} \quad \frac{B}{{m}^3} \quad \Rightarrow \quad B = {Nm}^3{C}^{-1} \] 
Step 3: Verifying the correct answer. The SI units of \( A \) and \( B \) are \( {Nm}^2{C}^{-1} \) and \( {Nm}^3{C}^{-1} \), respectively, which matches option (B).