Question

The electric flux is \( \varphi = \alpha \sigma + \beta \lambda \) where \( \lambda \) and \( \sigma \) are linear and surface charge density, respectively, and \( \left( \frac{\alpha}{\beta} \right) \) represents

• A. charge
• B. electric field
• C. displacement
• D. area

Hint:

Electric flux is a measure of the total electric field passing through a surface. The key idea is to understand how the surface charge density and linear charge density relate to the dimensions of the variables involved.

Answer 1

The Correct Option is C

To understand what \( \left( \frac{\alpha}{\beta} \right) \) represents in the equation for electric flux \( \varphi = \alpha \sigma + \beta \lambda \), we need to analyze the physical dimensions and properties involved.

Electric flux \( \varphi \) relates to the surface covered by an electric field and is typically calculated as the dot product of the electric field \( E \) and area \( A \), expressed as \( \varphi = E \cdot A \), implying that electric flux has dimensions of an electric field times an area.

In the given equation \( \varphi = \alpha \sigma + \beta \lambda \):

• \( \sigma \) is the surface charge density, with dimensions of charge per unit area \(([Q][L]^{-2})\).
• \( \lambda \) is the linear charge density, with dimensions of charge per unit length \(([Q][L]^{-1})\).

The coefficients \( \alpha \) and \( \beta \) must have dimensions that, when multiplied by their respective charge densities, result in the dimensions of electric flux \([E][A]\).

Let's analyze:

For the term \( \alpha \sigma \):

• \([E][A] = [\alpha][Q][L]^{-2}\)
• \([\alpha] = [E][L]^{2}/[Q]\)

For the term \( \beta \lambda \):

• \([E][A] = [\beta][Q][L]^{-1}\)
• \([\beta] = [E][L]/[Q]\)

Now, compute the ratio \( \frac{\alpha}{\beta} \):

• \(\frac{\alpha}{\beta} = \frac{[E][L]^{2}/[Q]}{[E][L]/[Q]} = [L]\)

Thus, the ratio \( \frac{\alpha}{\beta} \) represents a physical quantity with dimensions of length, typically associated with displacement in physics. Therefore, the correct interpretation of \( \left( \frac{\alpha}{\beta} \right) \) in this context is displacement.