Ch. 29 - Electromagnetic Induction and Faraday's Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 29 - Electromagnetic Induction and Faraday's Law

Problem 85b

Chapter 28, Problem 85b

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.

Verified step by step guidance

Step 1: Understand the relationship between the induced voltage (V) and the speed of the cart (υ). According to Faraday's law of electromagnetic induction, the induced voltage is proportional to the rate of change of magnetic flux, which in this case depends on the speed of the cart moving through the magnetic field. The theoretical slope of the graph is given by the product of the magnetic field strength (B) and the effective length of the coil (L).

Step 2: Analyze the experimental graph. The slope of the experimental graph can be determined by performing a linear regression or calculating the change in voltage (ΔV) divided by the change in speed (Δυ) from the data points provided in the table.

Step 3: Compare the experimental slope to the theoretical slope. The theoretical slope is calculated using the formula:

slope = B \times L

. Ensure you have the values for B (magnetic field strength) and L (effective length of the coil) from the problem or experiment setup.

Step 4: Calculate the percentage error using the formula:

% error = \frac{| Experimental slope - Theoretical slope |}{Theoretical slope} \times 100

. Substitute the values of the experimental slope and theoretical slope into this formula.

Step 5: Interpret the result. The percentage error quantifies the deviation of the experimental slope from the theoretical slope. A small percentage error indicates good agreement between the experimental and theoretical values, while a larger percentage error suggests potential experimental inaccuracies or assumptions that need to be revisited.

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Key Concepts

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

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit. The induced voltage is directly proportional to the rate of change of the magnetic flux and the number of turns in the coil. This principle is fundamental in understanding how the motion of the cart through the magnetic field generates voltage.

Slope of a Graph in Physics

In physics, the slope of a graph represents the relationship between two variables. For instance, in the context of induced voltage versus speed, the slope indicates how much the induced voltage changes with respect to the speed of the cart. Analyzing the slope allows for the comparison of experimental results with theoretical predictions, which is essential for calculating the percentage error.

Percentage Error Calculation

Percentage error is a way to express the accuracy of a measurement by comparing the difference between the experimental value and the theoretical value to the theoretical value itself. It is calculated using the formula: % error = |(experimental value - theoretical value) / theoretical value| × 100%. This concept is crucial for evaluating the reliability of experimental results in relation to established theories.

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Textbook Question

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

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:

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.

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