Find the dimension of `C/V` where C= Capacitance and V is potential difference

Answer

To find the dimension of \( \frac{C}{V} \) where \( C \) is capacitance and \( V \) is potential difference, we can follow these steps:

### Step 1: Understand the relationship between charge, capacitance, and potential difference
From the definition of capacitance, we know that:
\[ Q = CV \]
where \( Q \) is the charge, \( C \) is the capacitance, and \( V \) is the potential difference.

### Step 2: Rearranging the equation
We can rearrange this equation to express capacitance \( C \):
\[ C = \frac{Q}{V} \]

### Step 3: Substitute the expression for potential difference
The potential difference \( V \) can be expressed in terms of work \( W \) and charge \( Q \):
\[ V = \frac{W}{Q} \]
Substituting this into the expression for capacitance gives:
\[ C = \frac{Q}{\frac{W}{Q}} = \frac{Q^2}{W} \]

### Step 4: Find the expression for \( \frac{C}{V} \)
Now, we can find \( \frac{C}{V} \):
\[ \frac{C}{V} = \frac{Q^2/W}{W/Q} = \frac{Q^2}{W} \cdot \frac{Q}{W} = \frac{Q^3}{W^2} \]

### Step 5: Determine the dimensions of charge and work
The dimension of charge \( Q \) is:
\[ [Q] = [A][T] = A \cdot T \]
The dimension of work \( W \) is:
\[ [W] = [M][L^2][T^{-2}] = M \cdot L^2 \cdot T^{-2} \]

### Step 6: Substitute the dimensions into the expression
Now substituting these dimensions into our expression for \( \frac{C}{V} \):
\[ \frac{C}{V} = \frac{(A \cdot T)^3}{(M \cdot L^2 \cdot T^{-2})^2} \]
This simplifies to:
\[ \frac{C}{V} = \frac{A^3 \cdot T^3}{M^2 \cdot L^4 \cdot T^{-4}} = \frac{A^3 \cdot T^{3+4}}{M^2 \cdot L^4} = \frac{A^3 \cdot T^7}{M^2 \cdot L^4} \]

### Step 7: Final expression for dimensions
Thus, the final dimension of \( \frac{C}{V} \) is:
\[ \frac{C}{V} = M^{-2} \cdot L^{-4} \cdot T^7 \cdot A^3 \]

### Final Answer
The dimension of \( \frac{C}{V} \) is:
\[ [M^{-2} \cdot L^{-4} \cdot T^7 \cdot A^3] \]