You measure the magnitude of the external force F that must be applied to a rectangular conducti

step 1 Okay, so for Part A, it wants us to graph V, or sorry, F versus V. So first it wants us to talk

You measure the magnitude of the external force F that must...

You measure the magnitude of the external force $\overrightarrow{\boldsymbol{F}}$ that must be applied to a rectangular conducting loop to pull it at constant speed $v$ out of a region of uniform magnetic field $\vec{B}$ that is directed out of the plane of Fig. P29.66. The loop has dimensions $14.0 \mathrm{~cm}$ by $8.00 \mathrm{~cm}$ and resistance $4.00 \times 10^{-3} \Omega ;$ it does not change shape as it moves. The measurements you collect are listed in the table. $$\begin{array}{l|lllll} \boldsymbol{F}(\mathbf{N}) & 0.10 & 0.21 & 0.31 & 0.41 & 0.52 \\ \hline \boldsymbol{v}(\mathrm{cm} / \mathrm{s}) & 2.0 & 4.0 & 6.0 & 8.0 & 10.0 \end{array}$$ (a) Plot the data as a graph of $F$ versus $v$. Explain why the data points plotted this way lie close to a straight line, and determine the slope of the best-fit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the uniform magnetic field. (c) In Fig. $\mathrm{P} 29.66,$ is the current induced in the loop clockwise or counterclockwise? (d) At what rate is electrical energy being dissipated in the loop when the speed of the loop is $5.00 \mathrm{~cm} / \mathrm{s} ?$

You measure the magnitude of the external force $\overrightarrow{\boldsymbol{F}}$ that must be applied to a rectangular conducting loop to pull it at constant speed $v$ out of a region of uniform magnetic field $\vec{B}$ that is directed out of the plane of Fig. P29.66. The loop has dimensions $14.0 \mathrm{~cm}$ by $8.00 \mathrm{~cm}$ and resistance $4.00 \times 10^{-3} \Omega ;$ it does not change shape as it moves. The measurements you collect are listed in the table.
$$\begin{array}{l|lllll}
\boldsymbol{F}(\mathbf{N}) & 0.10 & 0.21 & 0.31 & 0.41 & 0.52 \\
\hline \boldsymbol{v}(\mathrm{cm} / \mathrm{s}) & 2.0 & 4.0 & 6.0 & 8.0 & 10.0
\end{array}$$
(a) Plot the data as a graph of $F$ versus $v$. Explain why the data points plotted this way lie close to a straight line, and determine the slope of the best-fit straight line for the data. (b) Use your graph from part
(a) to calculate the magnitude $B$ of the uniform magnetic field. (c) In Fig. $\mathrm{P} 29.66,$ is the current induced in the loop clockwise or counterclockwise? (d) At what rate is electrical energy being dissipated in the loop when the speed of the loop is $5.00 \mathrm{~cm} / \mathrm{s} ?$

Key Concepts

Graphical Analysis of Proportional Relationships

Plotting physical quantities that are directly proportional to each other, such as force versus velocity in this context, is a useful technique in physics. A straight-line graph indicates a linear relationship where the slope represents a combination of constants from the underlying theory, allowing experimental determination of parameters like the magnetic field strength.

Faraday's Law of Electromagnetic Induction

This principle states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in the loop. It is a fundamental concept that connects magnetic fields with induced electric currents and explains how electric generators and many electromagnetic devices operate.

Motional EMF

Motional EMF arises when a conductor moves through a magnetic field, causing the free charges in the conductor to experience a magnetic force. This results in a potential difference across the conductor that is proportional to the magnetic field strength, the length of the conductor within the field, and the speed at which it moves.

Lenz's Law

Lenz's Law provides the direction of the induced current by stating that the induced EMF and current always oppose the change in magnetic flux that produced them. This opposition ensures the conservation of energy and is key to understanding the behavior of systems with induced currents.

Lorentz Force on a Current-Carrying Conductor

The Lorentz force is the force experienced by a current-carrying conductor in a magnetic field. The magnitude of this force depends on the current in the conductor, the length of the conductor in the magnetic field, and the strength of the magnetic field, and it explains the mechanical forces acting on the moving loop.

Ohm's Law in Electric Circuits

Ohm's Law relates the current through a conductor to the voltage across it and its resistance (I = V/R). In contexts involving induced EMF, it is used to determine the magnitude of the induced current in a circuit based on the induced voltage and the circuit's resistance.

Joule Heating (Electrical Energy Dissipation)

Joule heating refers to the conversion of electrical energy into thermal energy through the resistance in an electric circuit. The rate of energy dissipation (power) is calculated as P = I²R, and it is an important consideration in systems where induced currents lead to energy losses.

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