You are conducting an experiment in which a metal bar of length 6.00 cm and mass 0.200 kg slid

You are conducting an experiment in which a metal bar of...

You are conducting an experiment in which a metal bar of length $6.00 \mathrm{~cm}$ and mass $0.200 \mathrm{~kg}$ slides without friction on two parallel metal rails (Fig. P29.65). A resistor with resistance $R=0.800 \Omega$ is connected across one end of the rails so that the bar, rails, and resistor form a complete conducting path. The resistances of the rails and of the bar are much less than $R$ and can be ignored. The entire apparatus is in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}$ that is directed into the plane of the figure. You give the bar an initial velocity $v=20.0 \mathrm{~cm} / \mathrm{s}$ to the right and then release it, so that the only force on the bar then is the force exerted by the magnetic field. Using high-speed photography, you measure the magnitude of the acceleration of the bar as a function of its speed. Your results are given in the table (next page): $$\begin{array}{l|rrrrrr} \boldsymbol{v}(\mathrm{cm} / \mathrm{s}) & 20.0 & 16.0 & 14.0 & 12.0 & 10.0 & 8.0 \\ \hline a\left(\mathrm{~cm} / \mathrm{s}^{2}\right) & 6.2 & 4.9 & 4.3 & 3.7 & 3.1 & 2.5 \end{array}$$ (a) Plot the data as a graph of $a$ versus $v$. Explain why the data points plotted this way lie close to a straight line, and determine the slope of the bestfit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the magnetic field. (c) While the bar is moving, which end of the resistor, $a$ or $b$, is at higher potential? (d) How many seconds does it take the speed of the bar to decrease from $20.0 \mathrm{~cm} / \mathrm{s}$ to $10.0 \mathrm{~cm} / \mathrm{s} ?$

You are conducting an experiment in which a metal bar of length $6.00 \mathrm{~cm}$ and mass $0.200 \mathrm{~kg}$ slides without friction on two parallel metal rails (Fig. P29.65). A resistor with resistance $R=0.800 \Omega$ is connected across one end of the rails so that the bar, rails, and resistor form a complete conducting path. The resistances of the rails and of the bar are much less than $R$ and can be ignored. The entire apparatus is in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}$ that is directed into the plane of the figure. You give the bar an initial velocity $v=20.0 \mathrm{~cm} / \mathrm{s}$ to the right and then release it, so that the only force on the bar then is the force exerted by the magnetic field. Using high-speed photography, you measure the magnitude of the acceleration of the bar as a function of its speed. Your results are given in the table (next page): $$\begin{array}{l|rrrrrr}
\boldsymbol{v}(\mathrm{cm} / \mathrm{s}) & 20.0 & 16.0 & 14.0 & 12.0 & 10.0 & 8.0 \\
\hline a\left(\mathrm{~cm} / \mathrm{s}^{2}\right) & 6.2 & 4.9 & 4.3 & 3.7 & 3.1 & 2.5
\end{array}$$ (a) Plot the data as a graph of $a$ versus $v$. Explain why the data points plotted this way lie close to a straight line, and determine the slope of the bestfit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the magnetic field. (c) While the bar is moving, which end of the resistor, $a$ or $b$, is at higher potential? (d) How many seconds does it take the speed of the bar to decrease from $20.0 \mathrm{~cm} / \mathrm{s}$ to $10.0 \mathrm{~cm} / \mathrm{s} ?$

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic environment of a conductor induces an electromotive force (emf) within it. In experiments where conductors move in a magnetic field, this induced emf is directly related to the velocity of the conductor. This concept is foundational and is quantified by Faraday’s law, which states that the magnitude of the induced emf is proportional to the rate of change of the magnetic flux through the circuit.

Motional EMF

Motional emf refers to the voltage generated when a conductor moves through a magnetic field. This phenomenon is described by the equation emf = B?v, where B is the magnetic field strength, ? is the length of the conductor in the magnetic field, and v is the velocity perpendicular to the field. This induced voltage drives a current in the closed circuit and is a key effect in devices such as generators and electromagnetic damping setups.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. In the context of a current-carrying conductor, the force is given by F = I?B, where I is the current, ? is the effective length of the conductor, and B is the magnetic field strength. This force acts perpendicular to both the velocity of the bar and the magnetic field direction, and in such experiments it typically opposes the motion, leading to a form of magnetic damping.

Magnetic Damping

Magnetic damping occurs when the motion of a conductor through a magnetic field generates eddy currents or an induced current that, in turn, produces forces opposing the motion. This process converts kinetic energy into electrical energy and subsequently into thermal energy through resistive losses. The damping force is proportional to the velocity, leading to a linear relationship between deceleration and speed in such systems.

Ohm's Law and Resistive Circuits

Ohm’s law, which states that the current through a conductor between two points is directly proportional to the voltage across the two points (I = V/R), is essential for understanding the behavior of the induced current in the resistor. In experimental setups where the resistance of the circuit plays a major role, it governs how the induced emf is converted into a current and how this current subsequently generates a magnetic force that opposes the motion.

Data Analysis and Linear Relationships

In the context of the experiment, the acceleration produced by the Lorentz force is proportional to the speed of the moving bar because the induced emf (and hence the current) is proportional to speed. When this acceleration is plotted versus speed, the data form a straight line whose slope is related to fundamental constants and experimental parameters like the magnetic field strength, mass of the bar, and circuit resistance. This linear relationship is used to extract quantitative values of the underlying physical quantities.

Potential Difference and Lenz’s Law

Lenz’s law states that the direction of the induced current is such that its magnetic effect opposes the change in magnetic flux that produced it. This principle determines the polarity of the induced voltage across the resistor. In experimental setups, knowing the direction of the induced current helps identify which end of the resistor is at a higher potential, an insight that is crucial for correctly interpreting the circuit behavior and ensuring consistency with energy conservation.

Transcript

00:01 Okay, so for part a, a graph of a versus v should be a straight line having slope equal to bl squared divided by mr.

00:33 So if i draw out this graph here, this will be a, which is in centimeters per second squared.

01:02 And then this would be vs which is centimeters per second so this is seven six five four three two one zero and then down here it'll be five ten fifteen and twenty so to draw these dots here we have in between the five and the ten between three and two right at the 10 we are right above 3 in between the 10 and 15 we are almost to 4 at 15 we are about right there in between the 15 and 20 we are right there and then at the 20 we are right above the 6 you draw a straight line and then there's the graph for part b, we have bl squared divided by mr, and that's equal to the slope.

02:31 So we have b equals the square root of slope times mr divided by l squared.

02:41 So that is you just plug in what you know into the equation.

02:45 0 .3071 seconds to negative one power times 0 .200 kilograms times 0 .200 kilograms times 0 .800, divide that by 0 .0600 meters squared, and you get 3 .69 ts.

03:13 For part c, the current flows in a counterclockwise direction in the circuit.

03:35 So therefore, the charges lose potential.

03:46 Energy as they pass through the resistor r from a to b, which makes point a at a higher than b.

04:23 So for part d, we know that.

04:30 A is equal to bl squared divided by m r times v...

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