Question 359: Electrostats & Capacitance

Question

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

Accepted Answer

Let's analyze the given expression step by step.
We are given the electric flux:
$$\phi = \alpha \sigma + \beta \lambda$$
where:
$$\sigma$$ is a surface charge density with units of charge per unit area (C/mÃÂ²).
$$\lambda$$ is a linear charge density with units of charge per unit length (C/m).
For the equation to be dimensionally consistent, the two terms on the right must have the same units as the flux $$\phi$$.
For the term $$\alpha \sigma$$:
Since $$\sigma$$ has units $$	ext{C/m}^2$$, the constant $$\alpha$$ must have units that convert $$\sigma$$ into the same units as the flux.
For the term $$\beta \lambda$$:
Since $$\lambda$$ has units $$	ext{C/m}$$, the constant $$\beta$$ carries its own units to ensure the term matches the fluxÃ¢ÂÂs dimensions.
Since both terms add to give the flux, they must share the same overall dimension. Thus, the dimensions of $$\alpha \sigma$$ and $$\beta \lambda$$ must be equal:
$$[\alpha] \cdot \frac{Q}{L^2} = [\beta] \cdot \frac{Q}{L}$$
Here, $$Q$$ represents the unit of charge and $$L$$ represents a unit of length. Canceling the common factors, we find:
$$\frac{[\alpha]}{[\beta]} = L$$
That is, the ratio $$\frac{\alpha}{\beta}$$ has the dimension of length.
Now, letÃ¢ÂÂs match this with the given options:
Option A: Displacement (commonly measured as a length)
Option B: Charge (has the unit Coulomb)
Option C: Electric Field (has units of V/m or N/C)
Option D: Area (has units of length squared, $$L^2$$)
Since $$\frac{\alpha}{\beta}$$ has the dimensions of a length, it logically corresponds to a displacement.
Thus, the correct answer is:
Option A Ã¢ÂÂ displacement.

Answer

displacement

Answer

charge

Answer

electric field

Answer

area

Electrostats & Capacitance

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

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