To determine the time at which the hour hand of a clock points in the same direction as the electric field at the center of the dial, we first need to analyze the arrangement of the charges and the resulting electric field they create. The clock face has negative charges positioned at each numeral, from -q at 1 o'clock to -12q at 12 o'clock. Let's break this down step by step.
Understanding Electric Fields from Point Charges
Electric fields produced by point charges can be calculated using Coulomb's law. The electric field (E) due to a point charge (Q) at a distance (r) is given by:
E = k * |Q| / r²
where k is Coulomb's constant. The direction of the electric field is away from positive charges and towards negative charges. Since all charges in this scenario are negative, the electric field will point towards each charge.
Analyzing the Charges on the Clock Face
We have the following charges at their respective positions:
- 1 o'clock: -q
- 2 o'clock: -2q
- 3 o'clock: -3q
- 4 o'clock: -4q
- 5 o'clock: -5q
- 6 o'clock: -6q
- 7 o'clock: -7q
- 8 o'clock: -8q
- 9 o'clock: -9q
- 10 o'clock: -10q
- 11 o'clock: -11q
- 12 o'clock: -12q
Each charge contributes to the net electric field at the center of the clock. The electric field vectors from each charge will have both magnitude and direction, and we need to find the resultant vector.
Calculating the Resultant Electric Field
To find the net electric field at the center, we can consider the contributions from each charge. The symmetry of the clock face allows us to simplify the calculations. The charges are arranged in a circular pattern, and their contributions can be analyzed in terms of their angles.
For example, the charge at 1 o'clock (-q) will produce an electric field pointing towards it, which is directed at an angle of 30 degrees from the vertical (12 o'clock position). Similarly, the charge at 2 o'clock (-2q) will point at 60 degrees, and so on. The general pattern is that the electric field from each charge can be resolved into x and y components.
Finding the Direction of the Electric Field
The total electric field at the center can be calculated by summing the x and y components of the electric fields from all the charges. However, due to the increasing magnitude of the charges, the contributions will not cancel out completely. The charge at 12 o'clock (-12q) will have the largest effect, pulling the net electric field towards it.
As we calculate the resultant electric field, we find that it will point towards the lower half of the clock face, specifically towards the 6 o'clock position, since the charges increase in magnitude as we move clockwise.
Determining the Time on the Clock
Since the electric field at the center points towards the 6 o'clock position, we need to find when the hour hand of the clock points in that same direction. The hour hand points directly downwards at 6 o'clock. Therefore, the hour hand will align with the electric field at:
6:00
In summary, the hour hand of the clock will point in the same direction as the electric field at the center of the dial when it is 6:00. This conclusion is based on the arrangement of the negative charges and the resultant electric field they create at the center of the clock face.
