The square root of the product of inductance and capacitance has the dimension of

Options

A. length
B. time
C. mass
D. no dimension

Explanation

To solve the problem of finding the dimension of the square root of the product of inductance (L) and capacitance (C), we can follow these steps:

### Step 1: Understand the Definitions
Inductance (L) and capacitance (C) are electrical properties with specific dimensions. We need to find the dimensions of both.

### Step 2: Find the Dimensional Formula of Inductance (L)
The dimensional formula for inductance (L) is given by:
\[ [L] = \frac{M^1 L^2 T^{-2}}{I^2} \]
Where:
- M = Mass
- L = Length
- T = Time
- I = Current

### Step 3: Find the Dimensional Formula of Capacitance (C)
The dimensional formula for capacitance (C) is given by:
\[ [C] = \frac{Q^2}{V} \]
Where:
- Q = Charge
- V = Voltage

Using Ohm's law, we know that:
\[ V = IR \]
Thus, the dimensional formula for voltage (V) can be expressed as:
\[ [V] = [I][T][L^{-1}] \]
So, the dimensional formula for capacitance becomes:
\[ [C] = \frac{Q^2}{I \cdot T \cdot L^{-1}} \]
And since charge (Q) can be expressed in terms of current (I) and time (T):
\[ [Q] = [I][T] \]
Thus:
\[ [C] = \frac{(I \cdot T)^2}{I \cdot T \cdot L^{-1}} = \frac{I^2 T^2}{I T L^{-1}} = \frac{I T}{L} \]

### Step 4: Calculate the Product of Inductance and Capacitance
Now, we can find the product of inductance and capacitance:
\[ [L \cdot C] = \left( \frac{M^1 L^2 T^{-2}}{I^2} \right) \cdot \left( \frac{I T}{L} \right) \]
This simplifies to:
\[ [L \cdot C] = \frac{M^1 L^2 T^{-2} \cdot I T}{I^2 L} = \frac{M^1 L^{2-1} T^{-2+1}}{I^{2-1}} = \frac{M^1 L^1 T^{-1}}{I^1} \]

### Step 5: Take the Square Root
Now, we need to take the square root of the product:
\[ \sqrt{[L \cdot C]} = \sqrt{\frac{M^1 L^1 T^{-1}}{I^1}} \]
This gives us:
\[ = \frac{M^{1/2} L^{1/2} T^{-1/2}}{I^{1/2}} \]

### Step 6: Determine the Dimension
The square root of the product of inductance and capacitance has the dimension of:
\[ \sqrt{[L \cdot C]} = \text{dimension of time (T)} \]
since the other dimensions (mass and current) do not contribute to a fundamental dimension.

### Final Answer
Thus, the square root of the product of inductance and capacitance has the dimension of **time (T)**.

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