Find the dimensions for electric power.

Answer

To find the dimensions of electric power, we can follow these steps:

### Step 1: Understand the formula for electric power
Electric power (P) is defined as the product of voltage (V) and current (I):
\[ P = V \times I \]

### Step 2: Write the expression for voltage
Voltage (V) can be defined in terms of work done (W) and charge (Q):
\[ V = \frac{W}{Q} \]

### Step 3: Substitute voltage in the power formula
Substituting the expression for voltage into the power formula gives:
\[ P = \left(\frac{W}{Q}\right) \times I \]

### Step 4: Rewrite the power formula
This can be rewritten as:
\[ P = \frac{W \times I}{Q} \]

### Step 5: Substitute the dimensions of work, charge, and current
Now, we need to express the dimensions of work (W), charge (Q), and current (I):
- The dimension of work done (W) is given by:
\[ [W] = [M][L^2][T^{-2}] \]
- The dimension of charge (Q) is given by:
\[ [Q] = [I][T] \]
- The dimension of current (I) is:
\[ [I] = [I] \]

### Step 6: Substitute the dimensions into the power formula
Now substituting these dimensions into the power formula:
\[ P = \frac{[M][L^2][T^{-2}] \times [I]}{[I][T]} \]

### Step 7: Simplify the expression
Now, we can simplify the expression:
- The current (I) in the numerator and denominator cancels out:
\[ P = \frac{[M][L^2][T^{-2}]}{[T]} \]
- This simplifies to:
\[ P = [M][L^2][T^{-3}] \]

### Step 8: Final result
Thus, the dimensions of electric power are:
\[ [P] = [M][L^2][T^{-3}] \]

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