**Problem Statement**A circular conducting loop with a radius of 1.00 m and a small gap filled with a 10.0 Ω resistor is oriented in the xy-plane. If a magnetic field of 2.0 T, making an angle of 30° with the z-axis increases to 9.0 T, in 2.0 s, what is the magnitude of the current that will be caused to flow in the conductor?**Analysis**This problem involves electromagnetic induction, where a changing magnetic field induces an electromotive force (EMF) in a conducting loop. This induced EMF will create a current, which flows through the resistor in the loop. **Solution**To solve this, use Faraday’s Law of Induction, which states that the induced EMF in a loop is equal to the negative rate of change of magnetic flux through the loop. The formula can be expressed as:\[ \text{EMF} = -\frac{\Delta \Phi}{\Delta t} \]where $\Delta \Phi$ is the change in magnetic flux and $\Delta t$ is the change in time.Next, calculate the magnetic flux ($\Phi$), given by:\[ \Phi = B \cdot A \cdot \cos(\theta) \]where:- $B$ is the magnetic field strength,- $A$ is the area of the loop ($A = \pi r^2$, with $r$ being the radius of the loop),- $\theta$ is the angle between the magnetic field and the normal to the loop.**Step-by-step Calculations:**1. Calculate the area of the loop: \[ A = \pi (1.00 \, \text{m})^2 = \pi \, \text{m}^2 \]2. Determine the initial and final magnetic flux: \[ \Phi_{\text{initial}} = 2.0 \, \text{T} \cdot \pi \, \text{m}^2 \cdot \cos(30^\circ) \] \[ \Phi_{\text{final}} = 9.0 \, \text{T} \cdot \pi \, \text{m}^2 \cdot \cos(30^\circ) \]3. Calculate the change in magnetic flux

radius (r) = 1 m Resistance (R) = 10 ohm Initial magnetic field (B) = 2 T θ = 30ofinal magnetic…

A circular conducting loop with a radius of 1.00 m and a small gap filled with a 10.0 resistor is oriented in the xy-pane. If a magnetic field of 2.0 T, making an angle of 30° with the z-axis increases to 9.0 T, in 2.0 s, what is the magnitude of the current that will be caused to flow in the conductor?

A circular conducting loop with a radius of 1.00 m and a small gap filled with a 10.0 resistor is oriented in the xy-pane. If a magnetic field of 2.0 T, making an angle of 30° with the z-axis increases to 9.0 T, in 2.0 s, what is the magnitude of the current that will be caused to flow in the conductor?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

**Problem Statement**

A circular conducting loop with a radius of 1.00 m and a small gap filled with a 10.0 Ω resistor is oriented in the xy-plane. If a magnetic field of 2.0 T, making an angle of 30° with the z-axis increases to 9.0 T, in 2.0 s, what is the magnitude of the current that will be caused to flow in the conductor?

**Analysis**

This problem involves electromagnetic induction, where a changing magnetic field induces an electromotive force (EMF) in a conducting loop. This induced EMF will create a current, which flows through the resistor in the loop.

**Solution**

To solve this, use Faraday’s Law of Induction, which states that the induced EMF in a loop is equal to the negative rate of change of magnetic flux through the loop. The formula can be expressed as:

\[ \text{EMF} = -\frac{\Delta \Phi}{\Delta t} \]

where $\Delta \Phi$ is the change in magnetic flux and $\Delta t$ is the change in time.

Next, calculate the magnetic flux ($\Phi$), given by:

\[ \Phi = B \cdot A \cdot \cos(\theta) \]

where:
- $B$ is the magnetic field strength,
- $A$ is the area of the loop ($A = \pi r^2$, with $r$ being the radius of the loop),
- $\theta$ is the angle between the magnetic field and the normal to the loop.

**Step-by-step Calculations:**

1. Calculate the area of the loop:
\[ A = \pi (1.00 \, \text{m})^2 = \pi \, \text{m}^2 \]

2. Determine the initial and final magnetic flux:
\[ \Phi_{\text{initial}} = 2.0 \, \text{T} \cdot \pi \, \text{m}^2 \cdot \cos(30^\circ) \]
\[ \Phi_{\text{final}} = 9.0 \, \text{T} \cdot \pi \, \text{m}^2 \cdot \cos(30^\circ) \]

3. Calculate the change in magnetic flux

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