In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field B of a permanent magnet. The speed of the cart v and the induced voltage V were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data:

Make a graph of the induced voltage, V, vs. the speed, v. Determine a best-fit linear equation for the data. Theoretically, the relationship between V and v is given by V = BN𝓁𝓋 where N is the number of turns of the coil, B is the magnetic field, and ℓ is the average of the inside and outside widths of the coil. In the experiment, B = 0.126 T, N = 50, and ℓ = 0.0561 m.

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

Step 1: Plot the data points from the table on a graph with speed (v) on the x-axis and induced voltage (V) on the y-axis. Use the given values: v = [0.367, 0.379, 0.465, 0.623, 0.630] m/s and V = [0.128, 0.135, 0.164, 0.221, 0.222] V.

Step 2: Draw a best-fit line through the plotted points. Ensure the line represents the trend of the data as closely as possible. This line will help determine the linear relationship between V and v.

Step 3: Use the equation of a straight line, y = mx + c, where m is the slope and c is the y-intercept, to find the best-fit linear equation. Calculate the slope (m) by selecting two points on the best-fit line and using the formula m = (ΔV / Δv).

Step 4: Substitute the slope (m) and one of the data points into the equation y = mx + c to solve for the y-intercept (c). This will give the complete best-fit linear equation for the graph.

Step 5: Compare the experimentally determined linear equation with the theoretical equation V = BNℓv. Substitute the given values for B = 0.126 T, N = 50, and ℓ = 0.0561 m into the theoretical equation to verify consistency between the experimental and theoretical results.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) or voltage in the wire. This phenomenon is described by Faraday's Law, which states that the induced voltage is proportional to the rate of change of the magnetic flux through the loop. In this experiment, the movement of the coil through the magnetic field generates an induced voltage that can be measured.

Linear Relationship

A linear relationship between two variables indicates that as one variable changes, the other variable changes at a constant rate. In the context of this experiment, the induced voltage (V) is expected to vary linearly with the speed (v) of the cart, as described by the equation V = BNℓv. This means that plotting V against v should yield a straight line, allowing for the determination of the slope, which represents the product of the magnetic field strength, the number of turns in the coil, and the average width of the coil.

Graphing Data

Graphing data is a crucial step in visualizing the relationship between two variables. In this experiment, the collected data of induced voltage versus speed can be plotted on a graph to identify trends and relationships. By applying a best-fit linear equation to the plotted points, one can analyze the correlation and derive meaningful conclusions about the relationship between the speed of the cart and the induced voltage, facilitating further analysis and understanding of the underlying physics.

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In a certain region of space near Earth’s surface, a uniform horizontal magnetic field of magnitude B exists above a level defined to be y = 0. Below y = 0, the field abruptly becomes zero (Fig. 29–63). A vertical square wire loop has resistivity ρ, mass density ρ_m, diameter d, and side length ℓ. It is initially at rest with its lower horizontal side at y = 0 and is then allowed to fall under gravity, with its plane perpendicular to the direction of the magnetic field. (a) While the loop is still partially immersed in the magnetic field (as it falls into the zero-field region), determine the magnetic “drag” force that acts on it at the moment when its speed is υ. (b) Assume that the loop achieves a constant terminal velocity V_T before its upper horizontal side exits the field. Determine a formula for V_T. (c) If the loop is made of copper and B = 0.80 T, find V_T.

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Textbook Question

What is the energy dissipated as a function of time in a circular loop of 18 turns of wire having a radius of 10.0 cm and a resistance of 2.0 Ω if the plane of the loop is perpendicular to a magnetic field given by B(t) = B₀e⁻ᵗ/ʳ with B₀ = 0.50 T and τ = 0.10 s?

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Textbook Question

Apply Faraday’s law, in the form of Eq. 29–8, to show that the static electric field between the plates of a parallel-plate capacitor cannot drop abruptly to zero at the edges, but must, in fact, fringe. Use the path shown dashed in Fig. 29–61. [Hint: Assume the contrary: that there is no fringing. Show that this assumption leads to a contradiction.]

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Textbook Question

In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field B of a permanent magnet. The speed of the cart v and the induced voltage V were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data:

Find the % error between the slope of the experimental graph and the theoretical value for the slope.

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