An electric dipole is placed at an angle of `60^(@)` with an electric field of magnitude `4xx10^(5) NC^(-1)`, It experiencs a torque of `8 sqrt(3) Nm`. If length of dipole is 2 cm, determine the magnitude of either charge of the dipole.

Explanation

To solve the problem step by step, we will use the formula for torque experienced by an electric dipole in an electric field.

### Step 1: Write down the formula for torque
The torque (\(\tau\)) experienced by an electric dipole in an electric field is given by the formula:
\[ \tau = P \cdot E \cdot \sin(\theta) \]
where:
- \(P\) is the dipole moment,
- \(E\) is the electric field strength,
- \(\theta\) is the angle between the dipole moment and the electric field.

### Step 2: Substitute the known values into the torque formula
Given:
- \(\tau = 8\sqrt{3}\) Nm
- \(E = 4 \times 10^5\) N/C
- \(\theta = 60^\circ\)

We can substitute these values into the formula:
\[ 8\sqrt{3} = P \cdot (4 \times 10^5) \cdot \sin(60^\circ) \]

### Step 3: Calculate \(\sin(60^\circ)\)
The value of \(\sin(60^\circ)\) is:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

### Step 4: Substitute \(\sin(60^\circ)\) into the equation
Now, substituting \(\sin(60^\circ)\) into the torque equation:
\[ 8\sqrt{3} = P \cdot (4 \times 10^5) \cdot \left(\frac{\sqrt{3}}{2}\right) \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ 8\sqrt{3} = 2\sqrt{3} \cdot (4 \times 10^5) \cdot P \]
Dividing both sides by \(2\sqrt{3}\):
\[ 4 = 4 \times 10^5 \cdot P \]

### Step 6: Solve for \(P\)
Now, divide both sides by \(4 \times 10^5\):
\[ P = \frac{4}{4 \times 10^5} = \frac{1}{10^5} = 10^{-5} \text{ C m} \]

### Step 7: Relate dipole moment to charge
The dipole moment \(P\) is also related to the charge \(Q\) and the distance \(d\) between the charges:
\[ P = Q \cdot d \]
Given that the length of the dipole \(d = 2 \text{ cm} = 0.02 \text{ m}\), we can rearrange this to find \(Q\):
\[ Q = \frac{P}{d} \]

### Step 8: Substitute \(P\) and \(d\) into the equation
Substituting the values we have:
\[ Q = \frac{10^{-5}}{0.02} \]

### Step 9: Calculate \(Q\)
Calculating this gives:
\[ Q = \frac{10^{-5}}{0.02} = 5 \times 10^{-4} \text{ C} \]

### Final Answer
Thus, the magnitude of either charge of the dipole is:
\[ Q = 5 \times 10^{-4} \text{ C} \]

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