What is the dinensions of impedance ?

Options

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

Explanation

To find the dimensions of impedance, we start with the definition of impedance (Z), which is given by the ratio of voltage (V) to current (I):

\[ Z = \frac{V}{I} \]

### Step 1: Write the dimensional formula for voltage (V)
Voltage can be expressed in terms of electric field (E) and distance (r):

\[ V = E \cdot r \]

The electric field (E) can be defined as force (F) per unit charge (Q):

\[ E = \frac{F}{Q} \]

Thus, we can express voltage as:

\[ V = \frac{F}{Q} \cdot r \]

### Step 2: Substitute the dimensional formulas
The dimensional formula for force (F) is:

\[ [F] = [M][L][T^{-2}] \]

The dimensional formula for charge (Q) is:

\[ [Q] = [I][T] \]

Now substituting these into the equation for voltage:

\[ V = \frac{[M][L][T^{-2}]}{[I][T]} \cdot [L] \]

### Step 3: Simplify the dimensional formula for voltage
Substituting and simplifying:

\[ V = \frac{[M][L][T^{-2}]}{[I][T]} \cdot [L] = \frac{[M][L^2][T^{-2}]}{[I][T]} \]

This simplifies to:

\[ V = [M][L^2][I^{-1}][T^{-3}] \]

So, the dimensional formula for voltage is:

\[ [V] = [M][L^2][I^{-1}][T^{-3}] \]

### Step 4: Write the dimensional formula for current (I)
The dimensional formula for current is simply:

\[ [I] = [I] \]

### Step 5: Substitute the dimensional formulas into the impedance equation
Now substituting the dimensional formulas of voltage and current into the impedance formula:

\[ Z = \frac{V}{I} = \frac{[M][L^2][I^{-1}][T^{-3}]}{[I]} \]

### Step 6: Simplify the dimensional formula for impedance
This gives us:

\[ Z = [M][L^2][I^{-1}][T^{-3}] \cdot [I^{-1}] \]

Thus, we have:

\[ Z = [M][L^2][I^{-2}][T^{-3}] \]

### Conclusion
The dimensions of impedance (Z) are:

\[ [Z] = [M][L^2][I^{-2}][T^{-3}] \]

### Final Answer
The dimensional formula for impedance is:

\[ [Z] = [M][L^2][I^{-2}][T^{-3}] \]

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