Problem 1 - University Physics with Modern Physics

Question

A single loop of wire with an area of 0.0900 m\(^2\) is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. (a) What emf is induced in this loop? (b) If the loop has a resistance of 0.600 \(\Omega\), find the current induced in the loop.

Step-by-Step Solution

Answer

(a) 0.0171 V, (b) 0.0285 A

Step 1: Identify the Given Values

We have a loop of wire with an area, \( A \), of 0.0900 m\(^2\). The initial magnetic field, \( B_0 \), is 3.80 T. It is decreasing at a rate of 0.190 T/s. The resistance of the loop, \( R \), is 0.600 \( \Omega \).

Step 2: Apply Faraday's Law of Induction

According to Faraday's Law, the induced emf \( ( \varepsilon ) \) in the loop is given by \( \varepsilon = - \frac{d\Phi}{dt} \), where \( \Phi \) is the magnetic flux. \( \Phi = B \cdot A \), so \( \frac{d\Phi}{dt} = A \cdot \frac{dB}{dt} \).

Step 3: Calculate the Rate of Change of Magnetic Flux

Since \( \frac{dB}{dt} = -0.190 \text{ T/s} \) (the field is decreasing), and \( A = 0.0900 \text{ m}^2 \), we substitute these values to get \( \frac{d\Phi}{dt} = 0.0900 \cdot (-0.190) = -0.0171 \text{ Wb/s} \).

Step 4: Calculate the Induced EMF

The induced emf, \( \varepsilon \), is equal to the negative change in magnetic flux: \[ \varepsilon = -(-0.0171) = 0.0171 \text{ V} \].

Step 5: Apply Ohm's Law to Find Induced Current

Using Ohm's Law, \( I = \frac{\varepsilon}{R} \), we find the induced current, \( I = \frac{0.0171}{0.600} = 0.0285 \text{ A} \).

Step 6: Summarize the Results

Thus, the emf induced in the loop is 0.0171 V, and the induced current is 0.0285 A.

Key Concepts

Magnetic Flux

Magnetic flux is a core concept in understanding electromagnetic induction. It represents the amount of magnetic field passing through a given area. This area could be the surface of a wire loop, like the one in our example.

It's defined as the product of the magnetic field strength, denoted as \( B \), and the area \( A \) it penetrates through, perpendicular to the surface. Mathematically, magnetic flux \( \Phi \) can be expressed as:

• \( \Phi = B \cdot A \cdot \cos(\theta) \)

In our problem, the angle \( \theta \) is zero because the magnetic field is perpendicular to the loop. This simplifies the calculation to \( \Phi = B \cdot A \).

Understanding magnetic flux is crucial as it links to how magnetic fields change over time, leading to phenomena like induced emf in electrical circuits.

Ohm's Law

Ohm's Law is a fundamental principle in the field of electricity and electronics. It describes the relationship between voltage, current, and resistance in a circuit. The formula can be expressed as:

• \( V = I \cdot R \)

Here, \( V \) is the voltage across the conductor, \( I \) is the current flowing through it, and \( R \) is the resistance.

In our exercise, we utilize Ohm's Law to find the current induced in the loop. By rearranging the formula to \( I = \frac{V}{R} \), we can substitute the induced emf for the voltage (\( V \)) and the known resistance (\( R \)) to calculate the current.

Ohm's Law serves as a foundational tool for solving many electric circuit problems, helping to relate changes in voltage to changes in current and resistance.

Induced EMF

Induced electromotive force (emf) is a voltage generated by changing magnetic fields, as described by Faraday's Law of Induction. Faraday's Law states that the induced emf in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. This can be formulated as:

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

The negative sign follows from Lenz's Law, indicating that the induced emf creates a current whose magnetic field opposes the change in the original magnetic flux.

In our example, with a single loop of wire, the magnetic field decreases over time, resulting in a changing magnetic flux. This change induces an emf in the loop. The specific problem sets up how to compute this induced emf using the rate of change of the magnetic field and the area of the loop.

The concept of induced emf is essential in understanding how electrical generators and transformers operate, relying on the principle of electromagnetic induction.

Magnetic Field

A magnetic field is a vector field surrounding a magnetic material or a moving electric charge. It exerts forces on other nearby magnetic materials and moving charges. The intensity of a magnetic field is measured in teslas (T).

In the context of the exercise, the magnetic field initially has a strength of 3.80 T and is uniform. It is directly related to the generation of magnetic flux through the loop. The fact that this field is decreasing at a rate of 0.190 T/s is crucial as it leads to the generation of an induced emf around the loop.

Understanding how magnetic fields interact with conductive loops allows us to design and interpret devices that convert mechanical energy to electrical energy, like those harnessed in power plants and electric motors, showcasing the practical power of magnetic fields in everyday applications.