Z' the impedance has dimensions

Options

A. (a)`M^(0)L^(2)T^(-2)`
B. (b)`M^(1)L^(2)T^(-3)A^(-2)`
C. (c)`M^(1)L^(2)T^(-2)A^(-1)`
D. (d)`M^(1)L^(1)T^(-2)A^(2)`

Answer

To determine the dimensions of impedance \( Z \), we can follow these steps:

### Step 1: Understand the relationship between impedance, voltage, and current.
Impedance \( Z \) is defined in terms of voltage \( V \) and current \( I \) in an AC circuit. The formula is given by:
\[ Z = \frac{V}{I} \]

### Step 2: Identify the dimensions of voltage and current.
- The dimension of voltage \( V \) can be expressed in terms of fundamental quantities:
\[ [V] = [M^1 L^2 T^{-3}] \]
This is derived from the formula \( V = IR \) where \( R \) (resistance) has dimensions of \( [M^1 L^2 T^{-3} A^{-2}] \).

- The dimension of current \( I \) is:
\[ [I] = [A^1] \]

### Step 3: Substitute the dimensions into the impedance formula.
Now substituting the dimensions of voltage and current into the impedance formula:
\[ [Z] = \frac{[V]}{[I]} = \frac{[M^1 L^2 T^{-3}]}{[A^1]} \]

### Step 4: Simplify the expression for impedance.
This gives us:
\[ [Z] = [M^1 L^2 T^{-3} A^{-1}] \]

### Step 5: Write the final expression for the dimensions of impedance.
Thus, the dimensions of impedance \( Z \) can be expressed as:
\[ [Z] = [M^1 L^2 T^{-3} A^{-1}] \]

### Step 6: Check the options provided in the question.
From the options provided, we can identify which one matches our derived dimensions. The correct option will be:
\[ [M^1 L^2 T^{-3} A^{-1}] \]

### Conclusion:
The final answer for the dimensions of impedance \( Z \) is:
\[ [M^1 L^2 T^{-3} A^{-1}] \]