To determine the magnetic field \( B \) when the sphere is in rotational equilibrium, we need to analyze the forces and torques acting on the sphere due to the current in the coil. The setup involves a sphere of radius \( r \) and mass \( m \) with a single turn coil wound around it, positioned such that the plane of the coil is parallel to an inclined plane and lies in the equatorial plane of the sphere.

Understanding the Forces at Play

When the sphere is placed on an inclined plane, it experiences gravitational force acting downwards, which can be broken down into two components: one perpendicular to the inclined plane and one parallel to it. The gravitational force \( F_g \) can be expressed as:

• Perpendicular component: \( F_{\perp} = mg \cos(\theta) \)
• Parallel component: \( F_{\parallel} = mg \sin(\theta) \)

Here, \( \theta \) is the angle of inclination of the plane. For the sphere to be in rotational equilibrium, the torque produced by the magnetic force must balance the torque due to the gravitational force.

Magnetic Force and Torque

The magnetic force \( F_m \) acting on the coil can be calculated using the formula:

F_m = i \cdot L \cdot B

where \( i \) is the current in the coil, \( L \) is the length of the coil (which, for a single turn, is the circumference of the sphere, \( 2\pi r \)), and \( B \) is the magnetic field strength.

The torque \( \tau \) due to the magnetic force about the center of the sphere is given by:

τ = r \cdot F_m = r \cdot (i \cdot 2\pi r \cdot B)

Setting Up the Equilibrium Condition

For the sphere to remain in rotational equilibrium, the torque due to the magnetic force must equal the torque due to the gravitational force. The torque due to gravity can be expressed as:

τ_g = r \cdot F_{\parallel} = r \cdot (mg \sin(\theta))

Setting the two torques equal gives us:

r \cdot (i \cdot 2\pi r \cdot B) = r \cdot (mg \sin(\theta))

We can simplify this equation by canceling \( r \) from both sides (assuming \( r \neq 0 \)):

i \cdot 2\pi r \cdot B = mg \sin(\theta)

Solving for the Magnetic Field B

Now, we can isolate \( B \) to find its value:

B = \frac{mg \sin(\theta)}{i \cdot 2\pi r}

This equation gives us the magnetic field strength \( B \) required to maintain the sphere in rotational equilibrium on the inclined plane, considering the mass of the sphere, the angle of inclination, and the current flowing through the coil.

Final Thoughts

In summary, the magnetic field \( B \) is directly proportional to the gravitational force component acting parallel to the incline and inversely proportional to both the current in the coil and the radius of the sphere. This relationship highlights the balance between magnetic forces and gravitational forces in maintaining equilibrium in this system.