An electric motor has a 1500 -turn, 15.0 -cm-diameter...

An electric motor has a 1500 -turn, 15.0 -cm-diameter circular coil on its armature. Find the magnetic field needed to produce a maximum torque of $25.0 \mathrm{~N} \cdot \mathrm{m}$ when the coil current is $12.0 \mathrm{~A}$.

Key Concepts

Maximum Torque Condition

For a coil in a magnetic field, the maximum torque is achieved when the magnetic force is most effective, which occurs when sin? = 1 (? = 90°). This condition means that the plane of the coil is oriented such that it is perpendicular to the magnetic field, optimizing the interaction between the current and the magnetic field. This concept is essential in designing and analyzing electric motors to ensure they operate under conditions that maximize performance.

Magnetic Field Calculation in Electric Motors

This concept involves using the known relationship between the torque, current, number of turns, and area to solve for the magnetic field strength required to generate a specified torque. By rearranging the torque equation (B = ? / (N I A)) under the maximum torque condition, one can determine the necessary magnetic field, which is a key parameter in the design and function of electric motors.

Magnetic Torque on a Current Loop

This concept describes the torque experienced by a current-carrying loop when placed in a magnetic field. The torque (?) depends on the number of turns (N), the current (I), the area of the loop (A), the magnetic field strength (B), and the sine of the angle between the loop’s normal and the magnetic field direction. The relationship is expressed as ? = N I A B sin?, and it is fundamental in understanding how electric motors convert electrical energy into mechanical work.

Area of a Circular Loop

The area of a loop is crucial in determining the torque generated by a current-carrying coil. When the loop is circular, its area is calculated using the formula A = ?r², where r is the radius. This geometrical calculation connects the physical dimensions of the loop with electromagnetic effects, as a larger area can lead to a higher torque for a given current and magnetic field.

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