If a small sphere of mass m and charge q is hung from a silk thread at an angle `theta` with the surface of a vertical charged conducting plate, then for equilibrium of sphere, the surface charge density of the plate is

Options

A. `epsilon_(0)((mg)/(q))tan theta`
B. `epsilon_(0)((2mg)/(q))tan theta`
C. `epsilon_(0)(mgq)tan theta`
D. `epsilon_(0)((mg)/(3q))tan theta`

Answer

To solve the problem of finding the surface charge density (\(\sigma\)) of a vertical charged conducting plate when a small sphere of mass \(m\) and charge \(q\) is hung from a silk thread at an angle \(\theta\), we can follow these steps:

### Step 1: Identify the Forces Acting on the Sphere
The forces acting on the sphere include:
- The gravitational force (\(F_g\)) acting downwards: \(F_g = mg\)
- The tension (\(T\)) in the thread acting at an angle \(\theta\) with the vertical
- The electric force (\(F_e\)) due to the electric field (\(E\)) created by the charged plate: \(F_e = qE\)

### Step 2: Resolve the Tension into Components
The tension in the thread can be resolved into two components:
- The vertical component: \(T \cos \theta\)
- The horizontal component: \(T \sin \theta\)

### Step 3: Apply the Equilibrium Conditions
For the sphere to be in equilibrium, the net force in both the vertical and horizontal directions must be zero.

**Vertical Direction:**
\[ T \cos \theta = mg \quad (1) \]

**Horizontal Direction:**
\[ T \sin \theta = qE \quad (2) \]

### Step 4: Express the Electric Field in Terms of Surface Charge Density
The electric field (\(E\)) due to an infinite charged plate with surface charge density \(\sigma\) is given by:
\[ E = \frac{\sigma}{2\epsilon_0} \quad (3) \]

### Step 5: Substitute Equation (3) into Equation (2)
Substituting the expression for \(E\) from Equation (3) into Equation (2):
\[ T \sin \theta = q \left(\frac{\sigma}{2\epsilon_0}\right) \quad (4) \]

### Step 6: Solve for Tension \(T\)
From Equation (1), we can express \(T\):
\[ T = \frac{mg}{\cos \theta} \quad (5) \]

### Step 7: Substitute \(T\) from Equation (5) into Equation (4)
Substituting Equation (5) into Equation (4):
\[ \frac{mg}{\cos \theta} \sin \theta = q \left(\frac{\sigma}{2\epsilon_0}\right) \]

### Step 8: Rearrange to Solve for Surface Charge Density \(\sigma\)
Rearranging the equation gives:
\[ \sigma = \frac{2mg \tan \theta}{q} \epsilon_0 \quad (6) \]

### Final Answer
Thus, the surface charge density (\(\sigma\)) of the plate is:
\[ \sigma = \frac{2mg}{q} \epsilon_0 \tan \theta \]