Question

A point charge is moving in clockwise direction in a circle with constant speed. Consider the magnetic field produced by the charge at a point \(P\) (not centre of the circle) on the axis of the circle. (1) only the magnitude of the magnetic field remains constant (2) only the direction of the magnetic field remains constant (3) magnetic field is constant in both direction and magnitude (4) both magnitude and direction of magnetic field change

Short answer

The magnetic field is constant in both direction and magnitude. So, the answer is (3).

Step-by-step solution

Understand the Scenario

A point charge is moving in a circle with constant speed in a clockwise direction. We need to consider the characteristics of the magnetic field produced by this charge at a point P, which is on the axis of the circle but not at the center.

Location of Point P

Point P is located on the axis of the circle, meaning it is perpendicular to the plane of the circle formed by the charge's path. P is not at the center, so it lies at some distance away from the plane.

Apply Biot-Savart Law

The Biot-Savart Law states that the magnetic field (B) due to a small segment of current-carrying wire is given by \[ d\textbf{B} = \frac{\text{\textmu}_0}{4\text{\textpi}} \frac{I d\textbf{l} \times \textbf{r}}{r^3} \]. For a moving charge, this is modified as a current loop, providing a resultant magnetic field perpendicular to the plane of motion.

Direction of Resultant Magnetic Field

Because the charge is moving in a circle, the direction of the magnetic field above and below the plane will depend on the right-hand rule. This direction remains constant in cylindrical coordinates because the charge's path is uniform.

Magnitude of Magnetic Field

As the charge moves with constant speed, the current loop created has a constant current magnitude, producing a constant magnetic field strength at point P.

Conclusion

Since both the direction and the magnitude of the magnetic field do not change due to the constant speed and uniform circular motion of the charge, the magnetic field at point P is constant in both direction and magnitude.

Key concepts

Biot-Savart Law

The Biot-Savart Law helps us calculate the magnetic field produced by a moving charge (or current). It states that the magnetic field \(B\) due to an element of current-carrying wire is given by: \[ d\textbf{B} = \frac{\text{\textmu}_0}{4\text{\textpi}} \frac{I d\textbf{l} \times \textbf{r}}{r^3} \]. Here, \(I\) is the current, \(d\textbf{l}\) is a small segment of the wire, \(\textbf{r}\) is the distance from the segment to the point where we are calculating the field, and \(\text{\textmu}_0\) is the permeability of free space. This translates the complex motion of charges into understandable magnetic field vectors.
For a moving point charge, this formula can be adapted to predict the field it creates. Because the point charge continuously moves in a circle, it generates a current loop.
The magnetic field produced is perpendicular to the plane where the charge moves.
The result is a neat and predictable field, especially useful when studying uniform motion.

Uniform Circular Motion

Uniform circular motion describes the motion of a point charge moving in a circle at a constant speed. Since the speed is uniform, the charge completes each circular path in the same amount of time.
This is important for understanding the magnetic field because the consistency in movement translates into a consistent effect on the field.

If the motion is uniform, the current produced is steady. Imagine holding a small ball and moving it around in a perfect circle, keeping your pace the same.
Even though the ball changes direction, the speed remains the same, creating a predictably steady pattern.
Applying this to the charged particle, the field produced retains a similar uniformity, leading to a constant magnetic field strength.

Magnetic Field Direction

The direction of the magnetic field produced by a moving charge follows a well-defined rule known as the right-hand rule. For a charge moving in a circle, if you curl your right hand in the direction of the charge's motion, your thumb points in the direction of the magnetic field above the plane.
Below the plane, the direction is the opposite.

This directional behavior remains consistent thanks to the uniformity of the circular motion. As the charge moves in consistent circles, the magnetic field produced keeps a steady orientation relative to the axis of that circle.

We see that at point P (which is not at the center but on the axis), the direction of the magnetic field does not change, contributing to its overall constancy. This steady state is crucial for understanding magnetic systems and predicting their behavior in various applications.