The unit and dimensions of impedance in terms of charge Q are

Options

A. ohm, `ML^(2)T^(-2)Q^(-2)`
B. ohm` ML^(2)T^(-1)Q^(-2)`
C. ohm, `ML^(2)T^(-2)Q^(-1)`
D. ohm, `MLT^(-1)Q^(-1)`

Answer

To find the unit and dimensions of impedance in terms of charge \( Q \), we will follow these steps:

### Step 1: Understand the Definition of Impedance
Impedance (\( Z \)) in an AC circuit is defined as the total resistance of the circuit. Therefore, it has the same units as resistance.

### Step 2: Identify the Unit of Resistance
The unit of resistance is the ohm (\( \Omega \)).

### Step 3: Use Ohm's Law
According to Ohm's Law, resistance (\( R \)) is defined as:
\[ R = \frac{V}{I} \]
where \( V \) is the voltage (potential difference) and \( I \) is the current.

### Step 4: Express Current in Terms of Charge
Current (\( I \)) can be expressed in terms of charge (\( Q \)) as:
\[ I = \frac{Q}{t} \]
where \( t \) is the time.

### Step 5: Substitute Current in Ohm's Law
Substituting the expression for current into Ohm's Law gives:
\[ R = \frac{V}{\frac{Q}{t}} = \frac{V \cdot t}{Q} \]

### Step 6: Express Voltage in Terms of Work and Charge
Voltage (\( V \)) can be expressed as:
\[ V = \frac{W}{Q} \]
where \( W \) is the work done.

### Step 7: Substitute Voltage in the Resistance Formula
Now substituting the expression for voltage into the resistance formula:
\[ R = \frac{\left(\frac{W}{Q}\right) \cdot t}{Q} = \frac{W \cdot t}{Q^2} \]

### Step 8: Find the Dimensions of Work
The dimensional formula for work (\( W \)) is:
\[ W = F \cdot s = [M][L][T^{-2}] \cdot [L] = [M][L^2][T^{-2}] \]

### Step 9: Substitute the Dimensions of Work
Substituting the dimensions of work into the resistance formula:
\[ R = \frac{[M][L^2][T^{-2}] \cdot [T]}{[Q^2]} = \frac{[M][L^2][T^{-1}]}{[Q^2]} \]

### Step 10: Write the Dimensions of Impedance
Thus, the dimensions of impedance (\( Z \)) are:
\[ Z = [M][L^2][T^{-1}][Q^{-2}] \]

### Final Answer
The unit of impedance is ohm (\( \Omega \)), and the dimensions in terms of charge \( Q \) are:
\[ [M][L^2][T^{-1}][Q^{-2}] \]

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