Question

A motor has a single loop inside a magnetic field of magnitude \(0.87 \mathrm{~T}\). 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 \(170 \mathrm{~V}\).

Short answer

Answer: The maximum angular speed possible for this motor when connected to a source of emf providing \(170 \mathrm{\ V}\) is approximately \(0.65\ \mathrm{rad/s}\).

Step-by-step solution

Calculate the magnetic flux

To start, we must determine the magnetic flux \(\phi\) in the loop. The magnetic flux is given by the product of the magnetic field \(B\), the loop area \(A\), and the cosine of the angle between the magnetic field and the normal to the loop surface, which can be represented as: \(\phi = B \cdot A \cdot \cos{\theta}\) Since the motor has a single loop, we know that at its maximum angular speed, the magnetic field and area vector will be perpendicular to each other. Therefore, the angle between them will be \(90^\circ\), and the cosine of \(90^\circ\) is 0. So, the magnetic flux is given by: \(\phi = B \cdot A \cdot \cos{90^\circ} = 0.87\ (\mathrm{T}) \cdot 300\ (\mathrm{cm}^{2}) \cdot 0 = 0\ \mathrm{Wb}\)

Apply Faraday's Law

Now, we apply Faraday's Law, which states that the induced emf in the loop is equal to the rate at which the magnetic flux changes. Faraday's Law is given by: \(|ε| = N\ |\frac{d\phi}{dt}|\) Since the motor has only one loop, \(N = 1\). Also, we are given the source emf as \(170\ \mathrm{V}\) (it is assumed that this is equal to the induced emf). Therefore, we can rewrite Faraday's Law for our problem as: \(|ε| = |\frac{d\phi}{dt}|\)

Solve for the angular speed

We are now looking for the maximum angular speed ω of the motor. Recall the magnetic flux expression: \(\phi = B \cdot A \cdot \cos{\theta}\) We know that \(\theta = \omega t\). So: \(\phi = B \cdot A \cdot \cos{\omega t}\) We then find the derivative of \(\phi\) with respect to time \(t\): \(\frac{d\phi}{dt}=-\omega B A \sin{\omega t}\) Substitute the derivative back into Faraday's Law and solve for the angular velocity ω: \(|ε| = |-ω B A \sin{\omega t}|\) \(\frac{170 \mathrm{\ V}}{0.87\ (\mathrm{T})\cdot 300\ (\mathrm{cm}^2)} = \sin{\omega t}\) Evaluate the sine function to find the maximum angular speed: \(\omega = \frac{1}{t} \arcsin{\left(\frac{170\ \mathrm{V}}{0.87\ (\mathrm{T})\cdot 300\ (\mathrm{cm}^2)}\right)}\) Notice that to obtain the maximum value of \(\omega\), the sine function must be at its maximum value, which is 1. This occurs when the argument inside the arcsin function is 1, so: \(\frac{170\ \mathrm{V}}{0.87\ (\mathrm{T})\cdot 300\ (\mathrm{cm}^2)} = 1\) Finally, solve for \(\omega\): \(\omega = \frac{170\ \mathrm{V}}{0.87\ (\mathrm{T})\cdot 300\ (\mathrm{cm}^2)}\) \(\omega \approx 0.65\ \mathrm{rad/s}\)

Answer

The maximum angular speed possible for this motor when connected to a source of emf providing \(170 \mathrm{\ V}\) is approximately \(0.65\ \mathrm{rad/s}\).

Key concepts

Faraday's Law

Understanding Faraday's Law is crucial in the realm of electromagnetism, especially when it comes to explaining how electric motors and generators work. Essentially, Faraday's Law of Electromagnetic Induction states that a change in magnetic environment of a coil of wire will induce an electromotive force (emf) in the coil. The amount of induced emf is directly proportional to the rate at which the magnetic flux through the coil is changing.

Faraday's Law can be mathematically expressed as:
\[|\varepsilon| = N \left|\frac{d\phi}{dt}\right|\] where \( \varepsilon \) is the induced emf, \( N \) is the number of turns in the wire coil, and \( \frac{d\phi}{dt} \) represents the rate of change of magnetic flux \( \phi \). In our example of the motor, with only one loop, the number of turns \( N \) is equal to 1, simplifying our calculations.

For students who might find direct application difficult, it's helpful to visualize the concept: imagine the loop as a hoop catching magnetic field lines; when these lines are 'cut' by the spinning hoop, electricity is produced.

Angular Speed

Angular speed, denoted as \( \omega \), is a measure of how quickly an object rotates or revolves relative to another point, in this case, how fast the motor's loop spins. It's a vector quantity, which means it has both a magnitude and a direction.

The angular speed is related to the linear speed of points on the rotating object and is given by the relationship:
\[\omega = \frac{v}{r}\]

Understanding it in terms of the loop in a motor:

Reflect on a carousel: the further you are from the center, the faster you have to move to complete one circle in the same amount of time. In the case of the motor, we determine angular speed to find at which rate the loop must spin to produce the required emf. Here, the challenge can be in relating angular speed to the change in flux, but linking it back to the movement of the loop through the magnetic field simplifies the understanding.

Induced EMF

Induced emf is the voltage generated across a conductor when it is exposed to a varying magnetic field. It's the cornerstone of how generators convert mechanical energy to electrical energy and vice versa for motors. In the context of our exercise, the induced emf is what drives the motor, and it arises due to the motor's loop cutting through magnetic field lines at a certain rate. The formula for induced emf as derived from Faraday's Law is given when there's a change in magnetic flux.

In simpler terms, if you think of the magnetic field as a river and the motor's loop as a paddle, the quicker you move the paddle through the water, the more force you feel; similarly, the faster you change the magnetic field around a loop (by spinning it), the higher the emf induced.

Faraday's Law is critical in understanding that the emf induced and how it relates to the device's operation. In our case, the maximum induced emf matches the external emf when the loop spins at the correct speed, thus allowing for the motor to function optimally at the given conditions.