A coil consists of 120 circular loops of wire of radius 4.8 cm . A current of 0.49 A runs throu

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A coil consists of 120 circular loops of wire of radius 4.8...

A coil consists of 120 circular loops of wire of radius $4.8 \mathrm{~cm} .$ A current of 0.49 A runs through the coil, which is oriented vertically and is free to rotate about a vertical axis (parallel to the $z$ -axis). It experiences a uniform horizontal magnetic field in the positive $x$ -direction. When the coil is oriented parallel to the $x$ -axis, a force of $1.2 \mathrm{~N}$ applied to the edge of the coil in the positive $y$ -direction can keep it from rotating. Calculate the strength of the magnetic field.

Key Concepts

Rotational Equilibrium

In a system where torques are acting, such as a magnetic dipole in a field, rotational equilibrium is achieved when the applied external torque exactly balances the magnetic torque. This balance of torques is a key idea in analyzing the steady state or static configurations of systems under the influence of magnetic forces.

Area of a Circular Loop

The area of a circular loop is a geometric property that is crucial in the calculation of a coil's magnetic dipole moment. Since the magnetic moment is proportional to the area enclosed by the current, knowing how to compute the area of a circle (using the formula A = ?r²) is an essential step in linking the geometry of the loop with its magnetic properties.

Magnetic Dipole Moment

A current loop (or a coil composed of several loops) has an associated magnetic dipole moment, which is a vector quantity that measures the strength and orientation of the magnetic source. It is calculated as the product of the current, the area of the loop, and the number of loops. This concept is central to understanding how current-carrying loops interact with external magnetic fields.

Torque on a Magnetic Dipole

When a magnetic dipole is placed in a magnetic field, it experiences a torque that tends to align the dipole with the field. The torque is given by the vector cross product of the magnetic moment and the magnetic field. This concept explains how a magnetic field can exert a rotational effect on the loop, which is fundamental to many electromagnetic devices.

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