Problem 6 - University Physics with Modern Physics

Question

A coil 4.00 \(\mathrm{cm}\) in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120 \mathrm{T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{T} / \mathrm{s}^{4}\right) t^{4} .\) 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 \(t=5.00 \mathrm{s} ?\)

Step-by-Step Solution

Answer

(a) The emf is \\(-0.03015 - 0.0003014 t^3\). (b) The current at \\(t=5\\) s is \\-0.000113\\) A.

Step 1: Understand Faraday's Law of Induction

To find the induced emf, we will apply Faraday's Law of Electromagnetic Induction. According to Faraday's Law, the induced emf \( \varepsilon \) in a coil with \( N \) turns is given by: \[ \varepsilon = -N \frac{d\Phi_B}{dt} \] where \( \Phi_B \) is the magnetic flux through one loop of the coil.

Step 2: Calculate the Magnetic Flux

The magnetic flux \( \Phi_B \) through one loop is given by: \[ \Phi_B = B \cdot A \] where \( A \) is the area of a single loop of the coil. The radius of the coil is \( 4.00 \) cm or \( 0.04 \) m, so \[ A = \pi r^2 = \pi (0.04)^2 = 5.024 \times 10^{-3} \text{ m}^2 \] Thus, the flux is: \[ \Phi_B = (0.0120 t + 3.00 \times 10^{-5} t^4) \times 5.024 \times 10^{-3} \] \[ \Phi_B = 6.029 \times 10^{-5} t + 1.507 \times 10^{-7} t^4 \]

Step 3: Differentiate the Magnetic Flux

Differentiate the expression for magnetic flux \( \Phi_B(t) \) with respect to time \( t \): \[ \frac{d\Phi_B}{dt} = \frac{d}{dt}(6.029 \times 10^{-5} t + 1.507 \times 10^{-7} t^4) \] \[ \frac{d\Phi_B}{dt} = 6.029 \times 10^{-5} + 4 \times 1.507 \times 10^{-7} t^3 \] \[ \frac{d\Phi_B}{dt} = 6.029 \times 10^{-5} + 6.028 \times 10^{-7} t^3 \]

Step 4: Calculate the Induced EMF

Substitute \( \frac{d\Phi_B}{dt} \) into the expression for the induced emf: \[ \varepsilon = -500 \, (6.029 \times 10^{-5} + 6.028 \times 10^{-7} t^3) \] \[ \varepsilon = -500 \times (6.029 \times 10^{-5} + 6.028 \times 10^{-7} t^3) \] \[ \varepsilon = -0.03015 - 0.0003014 t^3 \] \[ \varepsilon = -(0.03015 + 0.0003014 t^3) \]

Step 5: Calculate the Current in the Resistor

Use Ohm's Law to find the current \( I \) in the resistor, \( I = \frac{\varepsilon}{R} \). The resistor's resistance \( R \) is given as 600 \( \Omega \): For \( t = 5 \) s: \[ \varepsilon(5) = -(0.03015 + 0.0003014 \times 5^3) \] \[ \varepsilon(5) = -(0.03015 + 0.0003014 \times 125) \] \[ \varepsilon(5) = -(0.03015 + 0.037675) \] \[ \varepsilon(5) = -0.067825 \] Thus, \[ I = \frac{-0.067825}{600} = -0.000113 \text{ A} \] The negative sign indicates the direction of the current, which is typically opposite to the assumed direction.

Key Concepts

Induced emf

In the realm of electromagnetism, induced electromotive force (emf) is a significant concept that explains how electricity can be generated from a changing magnetic field. According to Faraday's Law of Induction, the induced emf in a coil with several turns is related to the rate of change of magnetic flux through the coil. Simply put, whenever there is a change in the magnetic environment of a coil of wire, it results in an induced current. This phenomenon is fundamentally explained by the formula:

• \( \varepsilon = -N \frac{d\Phi_B}{dt} \)

Here, \( \varepsilon \) is the induced emf, \( N \) represents the number of turns in the coil, and \( \frac{d\Phi_B}{dt} \) is the rate of change of the magnetic flux.
In practice, when you have a coil placed in a magnetic field that varies over time, the change in magnetic field strength, even if marginal, will cause an emf to be induced. This is a cornerstone principle in the operation of devices like electric generators and transformers. Faraday's Law was the dawn of understanding how mechanical energy can be converted into electrical energy, an idea that underpins much of today's power generation technologies.

Magnetic flux

Magnetic flux (\( \Phi_B \)) is a measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. You could think of it as the total magnetic 'field lines' passing perpendicularly through a given area. The greater the number of lines passing through the coil, the greater the magnetic flux. Mathematically, it is given by:

• \( \Phi_B = B \cdot A \)

Where \( B \) is the magnetic field strength and \( A \) is the area of the loop.
For example, if a coil with a certain radius is in a magnetic field, its area can be calculated using the formula for the area of a circle, \( A = \pi r^2 \). In the given exercise, substituting the coil's radius into this will help us determine how much magnetic field permeates the coil.
Understanding magnetic flux is crucial because it connects the physical size and shape of the coil with the magnetic environment it is in. By knowing the magnetic flux, one can predict how strong the induced emf will be when there are adjustments in the magnetic field, thanks to the quoted expression in Faraday’s Law.

Ohm's Law

Ohm’s Law is a fundamental principle in the analysis of electrical circuits. It connects three essential quantities: voltage (or emf), current, and resistance, providing a straightforward relationship between them. This law states that the current (\( I \)) flowing 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. It is articulated as:

• \( I = \frac{V}{R} \)

Within the context of the given problem, once the induced emf (which acts like a voltage) is known, Ohm's Law enables us to calculate the current passing through a resistor connected to the coil. The resistance of the resistor is given as 600 ohms.
For instance, at a specific time, by substituting known values for the emf and resistance into Ohm's Law, the current can be accurately determined. This utility of Ohm's Law simplifies calculations in circuits and is pivotal in designing electronic devices, ensuring that the right amount of current flows through various components to function correctly.