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

The expression for electric flux is given by: \(\varphi = \alpha \sigma + \beta \lambda\)

Definitions:

• \(\sigma\): Surface charge density (charge per unit area), typically in coulombs per square meter (\((\text{C/m}^2)\)).
• \(\lambda\): Linear charge density (charge per unit length), typically in coulombs per meter (\((\text{C/m})\)).
• \(\alpha\) and \(\beta\): Constants with units related to flux.

The objective is to interpret the ratio \(\left( \frac{\alpha}{\beta} \right)\).

Unit analysis:

• Electric flux (\(\varphi\)) units: volts-meter (\((\text{V}\cdot\text{m})\)) or newton-meter squared per coulomb (\((\text{N}\cdot\text{m}^2/\text{C})\)).
• Surface charge density (\(\sigma\)) units: \((\text{C/m}^2)\).
• Linear charge density (\(\lambda\)) units: \((\text{C/m})\).
• For the equation \(\alpha \sigma\) to have the units of flux, \(\alpha\) must have units of \((\text{V}\cdot\text{m}^3/\text{C})\).
• For the equation \(\beta \lambda\) to have the units of flux, \(\beta\) must have units of \((\text{V}\cdot\text{m}^2/\text{C})\).

The ratio \(\left( \frac{\alpha}{\beta} \right)\) has the following units:

\(\frac{\text{V}\cdot\text{m}^3/ \text{C}}{\text{V}\cdot\text{m}^2/ \text{C}} = \text{m}\)

Therefore, \(\left( \frac{\alpha}{\beta} \right)\) represents a length or displacement.

Conclusion:

• displacement