The dimensional formula for conductance is

Options

A. `[M^(-1) L^(-2) T^3 A^(-2)]`
B. `[M^1 L^2 T^(-3) A^2]`
C. `[M^(-1)L^(-2) T^(-3) A^2]`
D. `[M^(-1)L^(-2)T^3 A^2]`

Answer

To find the dimensional formula for conductance, we can follow these steps:

### Step 1: Understand the relationship between conductance and resistance
Conductance (G) is defined as the inverse of resistance (R). Therefore, we can write:
\[ G = \frac{1}{R} \]

### Step 2: Write the formula for resistance
Resistance (R) can be expressed in terms of voltage (V) and current (I) as:
\[ R = \frac{V}{I} \]

### Step 3: Substitute the expression for resistance into the conductance formula
Substituting the expression for resistance into the conductance formula gives us:
\[ G = \frac{I}{V} \]

### Step 4: Find the dimensional formula for voltage (V)
Voltage (V) is defined as work done (W) per unit charge (Q). Therefore:
\[ V = \frac{W}{Q} \]
The dimensional formula for work (W) is given by:
\[ W = \text{Force} \times \text{Displacement} = (MLT^{-2}) \times L = ML^2T^{-2} \]
The dimensional formula for charge (Q) is:
\[ Q = I \times T \]
So, the dimensional formula for voltage (V) becomes:
\[ V = \frac{ML^2T^{-2}}{IT} = ML^2T^{-3}I^{-1} \]

### Step 5: Substitute the dimensional formula for voltage into the conductance formula
Now we can substitute the dimensional formula for voltage back into the conductance formula:
\[ G = \frac{I}{V} = \frac{I}{ML^2T^{-3}I^{-1}} \]

### Step 6: Simplify the expression
This simplifies to:
\[ G = \frac{I^2}{ML^2T^{-3}} = \frac{I^2T^3}{ML^2} \]

### Step 7: Write the final dimensional formula for conductance
Thus, the dimensional formula for conductance (G) is:
\[ G = M^{-1}L^{-2}T^3I^2 \]

### Final Answer:
The dimensional formula for conductance is:
\[ [G] = M^{-1}L^{-2}T^3I^2 \]

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