The dimensions of capacitance are_____

Answer

To find the dimensions of capacitance, we will follow these steps:

### Step 1: Understand the formula for capacitance
The capacitance \( C \) is defined as the charge \( Q \) stored per unit voltage \( V \):
\[ C = \frac{Q}{V} \]

### Step 2: Determine the dimensions of charge \( Q \)
Charge \( Q \) can be expressed in terms of current \( I \) and time \( t \):
\[ Q = I \cdot t \]
The dimension of current \( I \) is given as:
\[ [I] = M^0 L^0 T^1 A^1 \]
The dimension of time \( t \) is:
\[ [T] = M^0 L^0 T^1 A^0 \]
Thus, the dimension of charge \( Q \) is:
\[ [Q] = [I][t] = (M^0 L^0 T^1 A^1)(M^0 L^0 T^0 A^0) = M^0 L^0 T^1 A^1 \]

### Step 3: Determine the dimensions of voltage \( V \)
Voltage \( V \) can be expressed in terms of electric field \( E \) and distance \( d \):
\[ V = E \cdot d \]
First, we find the dimension of electric field \( E \):
\[ E = \frac{F}{Q} \]
Where \( F \) is force. The dimension of force \( F \) is:
\[ [F] = M^1 L^1 T^{-2} \]
Now substituting the dimension of charge \( Q \):
\[ [E] = \frac{[F]}{[Q]} = \frac{M^1 L^1 T^{-2}}{M^0 L^0 T^1 A^1} = M^1 L^1 T^{-3} A^{-1} \]
Next, we find the dimension of distance \( d \):
\[ [d] = M^0 L^1 T^0 A^0 \]
Now we can find the dimension of voltage \( V \):
\[ [V] = [E][d] = (M^1 L^1 T^{-3} A^{-1})(M^0 L^1 T^0 A^0) = M^1 L^2 T^{-3} A^{-1} \]

### Step 4: Calculate the dimensions of capacitance \( C \)
Now we can substitute the dimensions of \( Q \) and \( V \) into the capacitance formula:
\[ [C] = \frac{[Q]}{[V]} = \frac{M^0 L^0 T^1 A^1}{M^1 L^2 T^{-3} A^{-1}} \]
This simplifies to:
\[ [C] = M^{-1} L^{-2} T^{4} A^{2} \]

### Final Answer
Thus, the dimensions of capacitance are:
\[ [C] = M^{-1} L^{-2} T^{4} A^{2} \]

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