A magnetic dipole of fixed moment \(m\) is moved slowly along the axis of a circular coil of radius a carrying a fixed current. How does the force on the dipole vary with its distance \(x\) from the centre of the coil if \(m\) always lies along the axis? At what distance, if any, has the force a maximum value?

Understanding the magnetic field generated by a circular coil is essential when studying magnetism and electromagnetism. Imagine a circular coil with a current flowing through it; the coil generates a magnetic field that has a specific pattern. This field is strongest at the center of the coil and diminishes as you move away.

For a coil with radius 'a' and a distance 'x' from the center along the axis of the coil, the magnetic field 'B' can be expressed as:
\[\begin{equation}B = \frac{\mu_0 I a^2}{2(x^2 + a^2)^\frac{3}{2}}\end{equation}\]Here, \(\mu_0\) stands for the permeability of free space, and 'I' is the current through the coil. This equation shows that the magnetic field decreases as the distance 'x' increases, which follows the concept that magnetic fields tend to weaken with distance.

Ampere's circuital law is a foundational concept in physics that provides a way to calculate the magnetic field around a closed loop due to an electric current. Ampere’s Law states that the integral of the magnetic field 'B' along a closed loop is proportional to the electric current 'I' passing through the loop.

The relationship is mathematically described by:
\[\begin{equation}\oint B \cdot dl = \mu_0 I\end{equation}\]This law is particularly useful for calculating the magnetic field in symmetrical situations, such as coils or straight wires. When dealing with the circular coil, Ampere's law allows us to find the magnetic field at any point on the axis by integrating over a circular path concentric with the coil.

The gradient of a magnetic field represents how the field changes with respect to displacement in space. It can be thought of as the slope or rate of change of the magnetic field. When we speak about the force on a magnetic dipole, we're actually talking about how the field's intensity varies from one place to another.

The force 'F' on our dipole moment 'm' is given by the product of the magnitude of the magnetic moment and the rate of change of the magnetic field with respect to distance 'x':
\[\begin{equation}F = m \frac{d B}{d x}\end{equation}\]This shows that the force will depend not just on the strength of the magnetic dipole and the magnetic field but also on how the field varies along the direction of the dipole's movement. In essence, where the gradient is steeper, meaning the magnetic field changes more abruptly, the force felt by the dipole moment is stronger.

Optimizing the force on a magnetic dipole involves finding the point at which this force is at a maximum value. This is akin to finding the peak of a hill - the spot where the incline is no longer increasing.

For our magnetic dipole, we search for the maximum value of the force along the axis of a circular coil. Mathematically, we find the maximum force by differentiating the force equation with respect to 'x' and setting this derivative to zero. This corresponds to finding when the rate of change of the force with respect to 'x' stops increasing.

Solving for x

Once we have the derivative equation, we solve for 'x' where the derivative is zero, this gives us the 'x' value(s) at which the force is maximized. Then, we can conclude that the force on the dipole has a maximum value at a certain distance 'x' from the coil center, where 'x' is defined by the aforementioned relationship.