Problem 6 - University Physics with Modern Physics

Question

A coil 4.00 cm in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B =\) (0.0120 T/s)\(t\) + (3.00 \(\times\) 10\(^{-5}\) T/s\(^4)t^4\). The coil is connected to a 600-\(\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time \(t =\) 5.00 s?

Step-by-Step Solution

Answer

(a) \(\varepsilon(t) = -2.512(0.0120 + 1.2 \times 10^{-4} t^3)\). (b) \(I = -1.131 \times 10^{-4}\) A at \(t = 5.00 \, \text{s}\).

Step 1: Identify the Variables

We are given a coil with a radius of 4.00 cm (converted to 0.04 m for calculations), 500 turns, and a magnetic field expressed as \(B(t) = (0.0120 \text{ T/s})t + (3.00 \times 10^{-5} \text{ T/s}^4)t^4\). The coil is also connected to a 600-\(\Omega\) resistor. We need to find the induced emf in the coil as a function of time and the current at \(t = 5.00 \text{ s}\).

Step 2: Calculate the Area of the Coil

The area \(A\) of the coil can be calculated using the formula for the area of a circle: \(A = \pi r^2\). With \(r = 0.04 \text{ m}\), the area is:\[A = \pi (0.04)^2 = 5.024 \times 10^{-3} \text{ m}^2.\]

Step 3: Determine the Induced EMF Using Faraday's Law

The induced emf \(\varepsilon\) in a coil is given by Faraday's Law: \(\varepsilon = -N \frac{d\Phi}{dt}\), where \(N\) is the number of turns, \(\Phi\) is the magnetic flux \(\Phi = B(t) \cdot A\). Thus:\[\varepsilon(t) = -N \frac{d}{dt}(B(t) \cdot A)\]\[\varepsilon(t) = -500 \times \frac{d}{dt}\left((0.0120t + 3.00 \times 10^{-5} t^4) \cdot 5.024 \times 10^{-3}\right).\]

Step 4: Differentiate the Magnetic Flux

To find the rate of change of magnetic flux \(\Phi(t)\), perform the differentiation with respect to \(t\):\[\frac{d\Phi}{dt} = \frac{d}{dt}\left((0.0120t + 3.00 \times 10^{-5} t^4) \cdot 5.024 \times 10^{-3} \right).\]Simplifying,\[\frac{d\Phi}{dt} = 5.024 \times 10^{-3} \left(0.0120 + 4 \times 3.00 \times 10^{-5} t^3\right) = 5.024 \times 10^{-3} (0.0120 + 1.2 \times 10^{-4} t^3).\]

Step 5: Calculate the Induced EMF Function

Substitute \(\frac{d\Phi}{dt}\) into the expression for the induced emf, resulting in:\[\varepsilon(t) = -500 \times 5.024 \times 10^{-3} (0.0120 + 1.2 \times 10^{-4} t^3).\]Simplifying, we get:\[\varepsilon(t) = -2.512(0.0120 + 1.2 \times 10^{-4} t^3).\]

Step 6: Calculate Current at \(t = 5.00 \, \text{s}\)

Using Ohm's Law, \(I = \frac{\varepsilon}{R}\), where \(R = 600\, \Omega\), first find \(\varepsilon(5)\):\[\varepsilon(5) = -2.512(0.0120 + 1.2 \times 10^{-4} \times 5^3).\]\(\varepsilon(5) = -2.512(0.0120 + 0.015) = -2.512 \times 0.027 = -0.06784 \, \text{V}.\)Hence,\[I = \frac{-0.06784}{600} = -1.131 \times 10^{-4} \, \text{A}.\]

Key Concepts

Faraday's Law

Faraday's Law outlines how a changing magnetic field induces an electromotive force (emf) in a coil, which is fundamental to various applications like power generation and electric circuits. Imagine a coil placed in a magnetic field. If this field changes over time, it will create an emf within the coil. In formula form, Faraday's Law is expressed as:

• \[ \varepsilon = -N \frac{d\Phi}{dt} \]

Here, \(\varepsilon\) represents the induced emf, \(N\) is the number of turns in the coil, and \(\frac{d\Phi}{dt}\) is the rate of change of magnetic flux through the coil.

This negative sign adheres to Lenz's Law, which states that the induced emf will generate a current that opposes the change in magnetic flux.

In the given exercise, by plugging in the magnetic field's changing formula and performing the calculations, we derived \(\varepsilon(t) = -2.512(0.0120 + 1.2 \times 10^{-4} t^3)\), demonstrating how the change in the magnetic field over time can influence the induced emf in the coil.

Magnetic Flux

Magnetic flux measures the quantity of the magnetic field passing through a given area, with its comprehension being crucial in understanding electromagnetic induction. The magnetic flux \(\Phi\) for a flat circle in a uniform magnetic field is defined by:

• \[ \Phi = B \cdot A \]

Where \(B\) is the magnetic field strength, and \(A\) is the area through which the magnetic field lines pass.

For a coil perpendicular to a magnetic field, as in our exercise, the flux is simplified and doesn't include any angle component because the angle between the magnetic field lines and the normal to the surface is zero.

In the exercise, with a magnetic field \(B(t)\) as a function of time and an area \(A\) derived using the formula for a circle, the magnetic flux \(\Phi(t) = B(t) \times A\). Differentiating \(\Phi\) with respect to time gives us the rate of change crucial for finding the induced emf.

Ohm's Law

Ohm's Law is a fundamental principle within electronics, relating voltage, current, and resistance within a circuit. It is essential when solving for current in circuits involving induced emfs. Ohm's Law is expressed as:

• \[ I = \frac{V}{R} \]

Where \(I\) is the current, \(V\) is the voltage, and \(R\) is the resistance.

When an emf is induced in a coil connected to a resistor, as in the exercise, you can determine the resulting current passing through the resistor using this law. By substituting the calculated induced emf (from Faraday's Law) into the formula, you can derive the current.

In the exercise, at time \(t = 5.00\) seconds, the emf was found to be \(-0.06784\) V. By applying Ohm's Law with a resistor of 600 \(\Omega\), the current is calculated as \(-1.131 \times 10^{-4}\, \text{A}\), showcasing how Ohm’s Law is pivotal in determining current flow from induced emf.