A clock face has charges `-q, -2q, ,.....-12q` fixed at the position of the corresponding numerals on the dial. The clock hands do not disturb the net field due to point charges. At what time does the hour hand point in the direction of the electric field at the centre of the dial.

Explanation

To solve the problem of determining at what time the hour hand of a clock points in the direction of the electric field at the center of the dial, we can follow these steps:

### Step 1: Understand the Setup
The clock face has charges at each numeral from 1 to 12, with the charges being -q, -2q, -3q, ..., up to -12q. The electric field at the center of the clock is influenced by these charges.

**Hint:** Visualize the clock face and the distribution of charges at each numeral.

### Step 2: Calculate the Electric Field Contribution
Each charge contributes to the electric field at the center. The electric field \( E \) due to a point charge is given by:
\[ E = \frac{k \cdot |q|}{r^2} \]
where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the center (which is the same for all charges).

**Hint:** Remember that the direction of the electric field due to negative charges is towards the charge.

### Step 3: Analyze the Contributions from Each Charge
The total electric field at the center is the vector sum of the electric fields due to all the charges. Since the charges are negative, the electric field vectors will point towards each charge.

**Hint:** Consider pairs of charges that are diametrically opposite to simplify the calculation.

### Step 4: Pairing Charges
For each pair of diametrically opposite charges, calculate the resultant electric field. For example, the pair (-q and -7q), (-2q and -8q), and so on. The net electric field from each pair can be calculated.

**Hint:** Use symmetry to find the resultant direction of the electric field.

### Step 5: Determine the Resultant Electric Field Direction
By symmetry, the resultant electric field will point towards the bisector of the angles formed by the vectors from the pairs of charges. Since there are 12 charges, the angle between each charge is \( 30^\circ \).

**Hint:** Identify the bisector between the pairs of charges to find the direction of the resultant electric field.

### Step 6: Identify the Time on the Clock
The resultant electric field points towards the bisector between the charges corresponding to 9 and 10. The hour hand of the clock will point in this direction at 9:30.

**Hint:** Visualize the clock face to confirm the position of the hour hand at 9:30.

### Conclusion
The hour hand points in the direction of the electric field at the center of the dial at **9:30**.

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