The dimensions of RC is equal to :

Options

A. time
B. inverse time
C. square of invers time
D. square of time

Answer

To determine the dimensions of the product \( RC \) where \( R \) is resistance and \( C \) is capacitance, we will analyze the dimensions of each component step by step.

### Step 1: Identify the dimensions of resistance \( R \)
Resistance \( R \) is defined as the ratio of voltage \( V \) to current \( I \):
\[ R = \frac{V}{I} \]
The SI unit of voltage \( V \) is the volt (V), and the SI unit of current \( I \) is the ampere (A).

The volt can be expressed in terms of base units:
\[ 1 \, \text{V} = 1 \, \text{W}/\text{A} = 1 \, \text{(J/C)}/\text{A} = 1 \, \text{(kg m}^2/\text{s}^2)/\text{C}/\text{A} \]
Thus, the dimensions of voltage \( V \) are:
\[ [V] = [M^1 L^2 T^{-3} I^{-1}] \]
Now, substituting this into the equation for resistance:
\[ [R] = \frac{[V]}{[I]} = \frac{[M^1 L^2 T^{-3} I^{-1}]}{[I]} = [M^1 L^2 T^{-3} I^{-2}] \]

### Step 2: Identify the dimensions of capacitance \( C \)
Capacitance \( C \) is defined as the ratio of charge \( Q \) to voltage \( V \):
\[ C = \frac{Q}{V} \]
The SI unit of charge \( Q \) is the coulomb (C).

The dimensions of charge \( Q \) can be expressed in terms of current and time:
\[ 1 \, \text{C} = 1 \, \text{A} \cdot \text{s} \]
Thus, the dimensions of charge \( Q \) are:
\[ [Q] = [I^1 T^1] \]
Now substituting this into the equation for capacitance:
\[ [C] = \frac{[Q]}{[V]} = \frac{[I^1 T^1]}{[M^1 L^2 T^{-3} I^{-1}]} = [M^0 L^0 T^4 I^2] \]

### Step 3: Calculate the dimensions of \( RC \)
Now, we can find the dimensions of the product \( RC \):
\[ [RC] = [R] \cdot [C] = [M^1 L^2 T^{-3} I^{-2}] \cdot [M^0 L^0 T^4 I^2] \]
Multiplying these together:
\[ [RC] = [M^{1+0} L^{2+0} T^{-3+4} I^{-2+2}] = [M^1 L^2 T^1 I^0] \]

### Step 4: Simplify the dimensions
Since \( I^0 = 1 \), we can simplify the dimensions:
\[ [RC] = [M^1 L^2 T^1] \]
This indicates that the dimensions of \( RC \) are:
\[ [RC] = [T] \]
Thus, the dimensions of \( RC \) are equivalent to time.

### Final Answer
The dimensions of \( RC \) are \( [T] \) (time).

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