91Ó°ÊÓ

Ohm's Law is an essential principle in electrical circuits that relates voltage, current, and resistance. This law states that the current \(I\) passing through a conductor between two points is directly proportional to the voltage \(V\) across the two points and inversely proportional to the resistance \(R\) of the conductor. Mathematically, it is expressed as:

• \( I = \frac{V}{R} \)

In the context of our motor problem, we applied Ohm's Law to determine the current flowing through the motor's coil. Given a 9.0-V battery and 28 \(\Omega\) resistance, the current is calculated as \( I = 0.3214 \, A \). Knowing the current is crucial as it determines the amount of torque produced by the motor, linking directly with how effectively the motor can perform its mechanical tasks.

Torque in the context of an electric motor relates to the turning ability of the motor's rotor. The formula for torque \(\tau\) generated by a current loop placed in a magnetic field is:

• \( \tau = n \times I \times A \times B \times \sin(\theta) \)

where:

• \( n \) is the number of turns in the coil,
• \( I \) is the current,
• \( A \) is the area of one turn of the coil,
• \( B \) is the magnetic field strength,
• \( \theta \) is the angle between the magnetic field and the normal to the coil.

To achieve maximum torque, the coil should be perpendicular to the magnetic field, i.e., \(\sin(\theta) = 1\). For our motor, substituting the given values results in a maximum torque of \(3.214 \times 10^{-4} \, N \cdot m\). Torque is a key factor in determining the motor's ability to perform work, such as turning a fan or propelling a vehicle.

Magnetic fields play a crucial role in the operation of electric motors. A magnetic field exerts a force on a current-carrying wire, which can cause the wire to move—an essential concept for generating torque. In our scenario, the magnetic field is specified as 0.020 T. The influence of the magnetic field is also directly related to the effectiveness of the motor, as described by the torque formula. Several key effects of a magnetic field in an electric motor include:

• Creating the Lorentz force that causes the rotor to spin.
• Influencing the magnitude of the torque produced.
• Affecting the efficiency of energy conversion from electrical to mechanical form.

Understanding these magnetic field effects allows engineers to design more efficient motors by optimizing the magnetic field direction and strength, the number of coil turns, and other factors that determine motor performance.