An indection coil has an impedance of `10 ohm`. When an AC signal of frecinency `1000 HZ` is applied to the coil, the voltage lends the curreat by `45^(circ)`. If the inductance of the coil is `(1)/(sqrt(p) xx q pi)` henry, then find `(pq)`.

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) given the inductance of the coil in the form \( \frac{1}{\sqrt{p} \cdot q \cdot \pi} \) henry. We will use the information about impedance, frequency, and the phase difference between voltage and current.

### Step-by-Step Solution:

1. **Understanding Impedance and Phase Angle:**
The impedance \( Z \) of an inductor is given by the formula:
\[ Z = \sqrt{R^2 + (X_L)^2} \]
where \( R \) is the resistance (which is 0 for an ideal inductor), and \( X_L \) is the inductive reactance given by:
\[ X_L = 2\pi f L \]
Here, \( f \) is the frequency and \( L \) is the inductance.

2. **Given Values:**
- Impedance \( Z = 10 \, \Omega \)
- Frequency \( f = 1000 \, \text{Hz} \)
- Phase angle \( \phi = 45^\circ \)

3. **Calculating Inductive Reactance:**
Since the voltage leads the current by \( 45^\circ \), we can use the relationship:
\[ \tan(\phi) = \frac{X_L}{R} \]
For \( \phi = 45^\circ \), \( \tan(45^\circ) = 1 \), which implies:
\[ X_L = R \]
Since \( R = 0 \) for an ideal inductor, we can assume \( X_L = 10 \, \Omega \).

4. **Finding Inductance:**
Now we can set up the equation for \( X_L \):
\[ X_L = 2\pi f L \]
Substituting the known values:
\[ 10 = 2\pi (1000) L \]
Rearranging for \( L \):
\[ L = \frac{10}{2\pi \cdot 1000} = \frac{10}{2000\pi} = \frac{1}{200\pi} \, \text{H} \]

5. **Relating Inductance to Given Form:**
We are given that:
\[ L = \frac{1}{\sqrt{p} \cdot q \cdot \pi} \]
Setting the two expressions for \( L \) equal:
\[ \frac{1}{200\pi} = \frac{1}{\sqrt{p} \cdot q \cdot \pi} \]
Cancelling \( \pi \) from both sides:
\[ \frac{1}{200} = \frac{1}{\sqrt{p} \cdot q} \]
This leads to:
\[ \sqrt{p} \cdot q = 200 \]

6. **Finding \( pq \):**
We can square both sides to eliminate the square root:
\[ p \cdot q^2 = 40000 \]
Since we have \( \sqrt{p} \cdot q = 200 \), we can express \( p \) in terms of \( q \):
\[ p = \frac{200^2}{q^2} = \frac{40000}{q^2} \]
Substituting this back into the equation gives:
\[ \frac{40000}{q^2} \cdot q^2 = 40000 \]
Thus, \( pq = 400 \).

### Final Answer:
The value of \( pq \) is \( 400 \).