You measure the magnitude of the external force F⃗ that must be applied to a rectangular conduct

step 1 Hi everyone in this problem it is given. Length of the copper cable is 1 .5 km, that is 1500 met

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

You measure the magnitude of the external force $\vec{F}$ 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. $\mathbf{P 2 9 . 6 6 .}$ 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. $$ \begin{array}{l|lllll} \boldsymbol{F}(\mathbf{N}) & 0.10 & 0.21 & 0.31 & 0.41 & 0.52 \\ \hline \boldsymbol{v}(\mathbf{c m} / \mathbf{s}) & 2.0 & 4.0 & 6.0 & 8.0 & 10.0 \end{array} $$ (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 $\vec{F}$ 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. $\mathbf{P 2 9 . 6 6 .}$ 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.
$$
\begin{array}{l|lllll}
\boldsymbol{F}(\mathbf{N}) & 0.10 & 0.21 & 0.31 & 0.41 & 0.52 \\
\hline \boldsymbol{v}(\mathbf{c m} / \mathbf{s}) & 2.0 & 4.0 & 6.0 & 8.0 & 10.0
\end{array}
$$
(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

Faraday's Law of Induction

This principle states that a changing magnetic flux through a closed circuit induces an electromotive force (EMF) in the circuit. When a conductor moves through a magnetic field, the area of the loop exposed to the field changes over time, leading to a varying flux and thus inducing an EMF. This law is fundamental in explaining how generators work, as well as in analyzing induced currents in moving conductors.

Motional EMF

Motional EMF is the voltage generated across a conductor due to its motion through a magnetic field. The effect arises because the magnetic force acts on the free charges in the conductor, separating them and creating a potential difference. This concept is crucial when analyzing systems where conductors move relative to a magnetic field, linking the speed of movement, the strength of the field, and the dimensions of the conductor.

Lorentz Force on a Conductor

The Lorentz force is the force exerted on charged particles when they move through a magnetic field, given by the cross product of the velocity and the magnetic field. In a current-carrying conductor, this force acts on the entire current distribution, resulting in a net mechanical force. This concept is important for understanding how external forces must sometimes be applied to counteract these forces in order to maintain a constant velocity.

Ohm's Law

Ohm's Law relates the current flowing through a conductor to the voltage across it and the resistance of the conductor. In the context of electromagnetic induction, the induced EMF drives a current according to Ohm's Law, meaning that the magnitude of the induced current is determined by the ratio of the EMF to the resistance. This relationship is key when calculating induced currents and analyzing circuits under dynamic conditions.

Joule Heating (Electrical Energy Dissipation)

Joule heating refers to the process by which electrical energy is converted into thermal energy in a resistor as current flows through it. The rate of energy dissipation is given by the product of the square of the current and the resistance. This concept is central to understanding how induced currents lead to energy loss in circuits, an effect that is important in both practical applications and fundamental studies of electromagnetic systems.

Lenz's Law

Lenz's Law provides the direction of the induced EMF and current such that the magnetic field produced by the induced current opposes the change in the original magnetic flux. This law is essential in determining the orientation of forces and currents in electromagnetic induction phenomena, ensuring that energy conservation is maintained in the system.

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