Problem 11 - College Physics

Question

A flat, square coil with 15 turns has sides of length 0.120 \(\mathrm{m}\) . The coil rotates in a magnetic field of 0.0250 \(\mathrm{T}\) (a) What is the angular velocity of the coil if the maximum emf produced is 20.0 \(\mathrm{mV}^{\prime} ?\) (Hint: Look at the motional emf induced across the ends of the segments of the coil.) (b) What is the average emf at this angular velocity?

Step-by-Step Solution

Answer

(a) Angular velocity is 37.04 rad/s. (b) Average emf is 0 V.

Step 1: Understand the Problem

We have a square coil with multiple turns, rotating in a magnetic field, producing a maximum emf. We need to find the angular velocity to produce a given maximum emf and then determine the average emf at that velocity.

Step 2: Formula for Maximum EMF

The maximum electromotive force (emf), also known as the peak emf (\(\epsilon_{max}\)), produced by a rotating coil in a magnetic field is given by the formula \[\epsilon_{max} = NAB\omega \sin(\theta)\]where \(N\) is the number of turns, \(A\) is the area of the coil, \(B\) is the magnetic field strength, and \(\omega\) is the angular velocity. Since the maximum value occurs when \(\sin(\theta) = 1\), the formula simplifies to \[\epsilon_{max} = NAB\omega\]."

Step 3: Calculate the Area of the Coil

The coil is square-shaped with each side of 0.120 m. Calculate the area as:\[A = 0.120 \, \text{m} \times 0.120 \, \text{m} = 0.0144 \, \text{m}^2\].

Step 4: Determine Angular Velocity

Rearrange the formula from Step 2 to solve for \(\omega\):\[\omega = \frac{\epsilon_{max}}{NAB}\].Substitute \(\epsilon_{max} = 20.0\, \text{mV} = 0.0200\, \text{V}\), \(N = 15\), \(A = 0.0144\, \text{m}^2\), and \(B = 0.0250\, \text{T}\) to find:\[\omega = \frac{0.0200}{15 \times 0.0144 \times 0.0250} = 37.04\, \text{rad/s}\].

Step 5: Calculate Average EMF

The average emf over a complete rotation is zero because the induced emf is sinusoidal, and the average value of a sine wave over a period is zero.

Key Concepts

Electromotive Force (EMF)

Electromotive Force (EMF), often denoted as \( \epsilon \), is essentially the voltage generated by a source such as a battery or a moving conductor in a magnetic field. In this problem, a rotating coil generates EMF when it cuts through magnetic field lines. EMF can be thought of as the potential difference which causes electrons to move, creating an electric current.

The formula for calculating the maximum EMF when a coil is rotating in a magnetic field is given by:

• \(\epsilon_{max} = NAB\omega\)

where:

• \(N\) is the number of turns in the coil.
• \(A\) is the area of the coil.
• \(B\) is the magnetic field strength.
• \(\omega\) is the angular velocity of the coil.

The maximum EMF occurs when the coil is perpendicular to the magnetic field lines, maximizing the number of field lines cut per unit time. This is often the scenario in alternating current (AC) generators and various electromagnetic applications.

Magnetic Field

A magnetic field surrounds a magnet or a moving electric charge, acting with a force on other nearby magnets or charges. In this exercise, the magnetic field is uniform and has a strength of 0.0250 Tesla (T).

The interaction between the magnetic field and the rotating coil is crucial because the field lines must be "cut" by the coil to induce EMF. A stronger magnetic field results in a higher potential EMF, since the cutting action causes more movement of electrons within the coil.

• Magnetic fields are depicted through field lines which show the direction and strength.
• Indeed, in this scenario, the coil rotates in, producing a changing magnetic flux.

Understanding how magnetic fields work is key to solving many problems involving electromagnetism, including the computation of force on a current-carrying conductor within a magnetic field.

Rotating Coil

A rotating coil in a magnetic field functions as the basis for many electrical devices, like the generator. As described in the problem, every rotation of the coil changes the angle at which it intercepts magnetic field lines, altering the magnetic flux through the coil.

The coil has 15 turns, making it more efficient in inducing EMF by multiplying the effect each loop has. The area of each loop is 0.0144 square meters, which determines how much magnetic field it can engage.

• Rotation speed, or angular velocity (\(\omega\)), is critical: Faster rotation speeds induce greater EMF.
• The number of turns and coil area enhance its susceptibility to magnetic interaction.

Applications of rotating coils in magnetic fields include transforming mechanical energy into electrical energy, with diverse applications from simple motors to complex power plants.

Physics Problem Solving

Physics problem solving can be immensely gratifying with a step-by-step approach that dissects the different components of a problem sequentially. With the problem of a rotating coil in a magnetic field, each detail offers insight into the underlying principles of electromagnetism.

Here is how you can approach such problems:

• **Identify the knowns and unknowns:** Start by listing all given data and what needs to be determined, as seen with quantities like the coil’s length, turns, and EMF.
• **Apply appropriate formulas:** Use relevant physics equations to relate variables, guided by known principles, such as calculating maximum EMF using \(\epsilon_{max} = NAB\omega\).
• **Substitute values carefully:** Replace symbols with actual values, ensuring consistent units (e.g., meters, Teslas, volts).
• **Analyze the results:** Interpret solutions in the context of the problem, as the final step reveals both maximum and average EMF, reflecting real-world phenomena.

By applying these principles, students can confidently navigate physics problems, harnessing a mix of conceptual understanding and practical application.