A wire of linear charge density `lambda` passes through a cuboid of length l, breadth b and height h(l>b>h) in such a manner that the flux through the cuboid is maximum. The position of the wire is now changed, so that the flux through the cuboid is minimum. The raito of maximum flux to minimum flux will be

Options

A. `(sqrt(l^(2) + b^(2) + h^(2)))/(h)`
B. `(sqrt(l^(2) + b^(2)))/(h)`
C. `(h)/(sqrt(l^(2) + b^(2)))`
D. `(l)/(sqrt(l^(2) + b^(2) + h^(2)))`

Answer

To solve the problem, we need to find the ratio of the maximum electric flux to the minimum electric flux through a cuboid when a wire with linear charge density \( \lambda \) passes through it.

### Step-by-Step Solution:

1. **Understanding Electric Flux**:
The electric flux \( \Phi \) through a closed surface is given by Gauss's law:
\[ \Phi = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( Q_{\text{enc}} \) is the charge enclosed by the surface and \( \epsilon_0 \) is the permittivity of free space.

2. **Maximum Flux Condition**:
To maximize the flux through the cuboid, the wire should pass through the body diagonal of the cuboid. The body diagonal \( d \) of a cuboid with dimensions \( l \), \( b \), and \( h \) is given by:
\[ d = \sqrt{l^2 + b^2 + h^2} \]
The total charge \( Q_{\text{max}} \) enclosed when the wire passes through the body diagonal is:
\[ Q_{\text{max}} = \lambda \cdot d = \lambda \cdot \sqrt{l^2 + b^2 + h^2} \]

3. **Minimum Flux Condition**:
To minimize the flux through the cuboid, the wire should pass through the cuboid such that it covers only the smallest dimension, which is \( h \) (since \( l > b > h \)). Thus, the total charge \( Q_{\text{min}} \) enclosed in this case is:
\[ Q_{\text{min}} = \lambda \cdot h \]

4. **Calculating the Fluxes**:
Using Gauss's law, the maximum and minimum fluxes can be expressed as:
\[ \Phi_{\text{max}} = \frac{Q_{\text{max}}}{\epsilon_0} = \frac{\lambda \cdot \sqrt{l^2 + b^2 + h^2}}{\epsilon_0} \]
\[ \Phi_{\text{min}} = \frac{Q_{\text{min}}}{\epsilon_0} = \frac{\lambda \cdot h}{\epsilon_0} \]

5. **Finding the Ratio of Maximum to Minimum Flux**:
The ratio of maximum flux to minimum flux is:
\[ \frac{\Phi_{\text{max}}}{\Phi_{\text{min}}} = \frac{\frac{\lambda \cdot \sqrt{l^2 + b^2 + h^2}}{\epsilon_0}}{\frac{\lambda \cdot h}{\epsilon_0}} = \frac{\sqrt{l^2 + b^2 + h^2}}{h} \]

6. **Final Result**:
Thus, the ratio of maximum flux to minimum flux is:
\[ \frac{\Phi_{\text{max}}}{\Phi_{\text{min}}} = \frac{\sqrt{l^2 + b^2 + h^2}}{h} \]