Question

A motor has a single loop inside a magnetic field of magnitude \(0.870 \mathrm{~T}\). If the area of the loop is \(300 . \mathrm{cm}^{2}\), find the maximum angular speed possible for this motor when connected to a source of emf providing \(170 .\) V.

Short answer

Answer: The maximum angular speed of the motor is approximately 6493.10 rad/s.

Step-by-step solution

Convert given values to SI Units if necessary.

The area given is in cm², we need to convert it to m². 1 cm² = 0.0001 m². Given area: 300 cm² = 300 * 0.0001 m² = 0.03 m².

Calculate the maximum emf (induced voltage) in the coil.

We can use the formula for the maximum emf (induced voltage) generated in a rotating coil, which is: Emf = B * A * w Where Emf is the induced voltage, B is the magnetic field strength, A is the area of the loop, and w is the maximum angular speed of the motor. We want to find w. We are given: Emf = 170 V B = 0.870 T A = 0.03 m²

Solve for the maximum angular speed.

Now, we can rearrange the formula to solve for the maximum angular speed (w). w = Emf / (B * A) Now, plug in the values: w = 170 V / (0.870 T * 0.03 m²) w ≈ 6493.10 rad/s

Present the final result.

The maximum angular speed of the motor when connected to a source of emf providing 170 V is approximately 6493.10 rad/s.

Key concepts

Magnetic Field

A magnetic field is a region of space where magnetic forces can be observed. It's symbolized by the letter B and measured in teslas (T). Imagine the area around a magnet where other magnets or ferromagnetic materials like iron will experience a force – that's the magnetic field.

In our exercise, we’re given a magnetic field strength of 0.870 T, which is a significant field strength. In the context of motors and generators, magnetic fields are fundamental for operation, as they interact with electric currents to produce forces and motion, leading to the concept of electromagnetic induction.

Induced EMF

Electromagnetic induction is the process that leads to induced electromotive force (emf), often just referred to as voltage. It occurs when a conductor, such as a wire coil, moves through a magnetic field or when the magnetic field around the conductor changes. This change can induce a current in the conductor.

In terms of our motor problem, an emf of 170 V is provided by an external source to the motor. However, in practical situations, this could be the induced emf generated by the coil rotation within the motor itself. The maximum induced emf equation given in the solution, Emf = B * A * w, relates the magnetic field (B), the area of the loop (A), and the coil's angular speed (w) to the emf.

Coil Rotation

The rotation of a coil in a motor involves its movement within a magnetic field, which is directly related to the concept of induced emf. As the coil turns, it can cut across magnetic field lines, thereby generating an emf according to Faraday’s law of electromagnetic induction.

In the given problem, we are solving for the maximum angular speed, symbolized by w (in radians per second, or rad/s), which is the speed at which the coil rotates. This speed directly affects the induced emf — faster rotation leads to greater emf, according to the equation used in the solution.

SI Units Conversion

Converting to and from SI units (International System of Units) is crucial in physics problems to ensure accuracy and consistency. The SI unit for area is square meters (m²), so converting from square centimeters (cm²) to square meters is a common step when working with such problems.

As done in Step 1 of the solution, we use the conversion factor where 1 cm² = 0.0001 m². Multiplying the given area, 300 cm², by this factor yields 0.03 m². It's important to carry out these conversions correctly, as using inconsistent units can lead to incorrect results.