Chage `Q` is distributed to two different metwllic spheres having radii `R`and `2R` such that both spheres having equal surface charge densityh. Then charge on larger sphere is

Options

A. `(4Q)/(5)`
B. `(Q)/(5)`
C. `(3Q)/(5)`
D. `(5Q)/(4)`

Explanation

To solve the problem step by step, we need to find the charge on the larger sphere given that both spheres have equal surface charge density.

### Step 1: Understand the problem
We have two metallic spheres with radii \( R \) and \( 2R \). Let the charge on the smaller sphere (radius \( R \)) be \( q \) and the charge on the larger sphere (radius \( 2R \)) be \( q' \). The total charge distributed on both spheres is \( Q \).

### Step 2: Write the equation for total charge
The total charge on both spheres can be expressed as:
\[ q + q' = Q \]

### Step 3: Write the expression for surface charge density
The surface charge density \( \sigma \) for a sphere is given by the formula:
\[ \sigma = \frac{\text{Charge}}{\text{Surface Area}} \]
The surface area of a sphere is \( 4\pi r^2 \).

For the smaller sphere (radius \( R \):
\[ \sigma = \frac{q}{4\pi R^2} \]

For the larger sphere (radius \( 2R \)):
\[ \sigma = \frac{q'}{4\pi (2R)^2} = \frac{q'}{16\pi R^2} \]

### Step 4: Set the surface charge densities equal
Since both spheres have equal surface charge densities, we can set the two expressions for \( \sigma \) equal to each other:
\[ \frac{q}{4\pi R^2} = \frac{q'}{16\pi R^2} \]

### Step 5: Simplify the equation
We can cancel \( 4\pi R^2 \) from both sides:
\[ q = \frac{q'}{4} \]
This implies:
\[ q' = 4q \]

### Step 6: Substitute \( q' \) back into the total charge equation
Now we substitute \( q' = 4q \) into the total charge equation:
\[ q + 4q = Q \]
\[ 5q = Q \]

### Step 7: Solve for \( q \)
From the equation \( 5q = Q \), we can solve for \( q \):
\[ q = \frac{Q}{5} \]

### Step 8: Find \( q' \)
Now we can find \( q' \):
\[ q' = 4q = 4 \left(\frac{Q}{5}\right) = \frac{4Q}{5} \]

### Conclusion
The charge on the larger sphere (radius \( 2R \)) is:
\[ q' = \frac{4Q}{5} \]