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

Options

A. length
B. mass
C. time
D. frequency

Answer

To solve the problem, we need to find the dimensions of the square root of the product of inductance (L) and capacitance (C). Let's break this down step by step.

### Step 1: Identify the dimensions of inductance (L) and capacitance (C)

- The dimension of inductance (L) is given by:
\[ [L] = \frac{M L^2}{T^2 I^2} \]
where \(M\) is mass, \(L\) is length, \(T\) is time, and \(I\) is electric current.

- The dimension of capacitance (C) is given by:
\[ [C] = \frac{T^2 I^2}{M L^2} \]

### Step 2: Calculate the product of inductance and capacitance

Now, we calculate the product \(L \cdot C\):
\[ [L] \cdot [C] = \left(\frac{M L^2}{T^2 I^2}\right) \cdot \left(\frac{T^2 I^2}{M L^2}\right) \]

### Step 3: Simplify the product

When we multiply these two dimensions, we can see that \(M\), \(L^2\), \(T^2\), and \(I^2\) will cancel out:
\[ [L] \cdot [C] = \frac{M L^2 T^2 I^2}{T^2 I^2 M L^2} = 1 \]

Thus, the product \(L \cdot C\) is dimensionless.

### Step 4: Find the square root of the product

Now, we take the square root of the product:
\[ \sqrt{L \cdot C} = \sqrt{1} = 1 \]

### Step 5: Determine the dimensions

Since the square root of the product of inductance and capacitance is dimensionless, we conclude that:
\[ \sqrt{L \cdot C} \text{ has the dimension of } T \text{ (time)} \]

### Final Answer

The square root of the product of inductance and capacitance has the dimension of **time (T)**.

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