Terminal speed A bar of length L=0.36 m is free to slide without friction on horizontal rails a

Terminal speed A bar of length L=0.36 m is free to slide...

Key Concepts

Motional Electromotive Force (emf)

This concept describes the generation of an electromotive force in a conductor moving through a magnetic field. The induced emf is proportional to the velocity of the conductor, the magnetic field strength, and the length of the conductor. It is key to understanding how motion in a magnetic environment results in electrical energy being produced, which in turn affects the current in the circuit.

Lorentz Force on a Current-Carrying Conductor

This principle explains that a conductor through which an electric current flows in the presence of a magnetic field experiences a force. The magnitude of this force is given by the product of the current, the length of the conductor in the magnetic field, and the magnetic field strength. It is critical for determining the acceleration of the moving bar as the magnetic force acts as the net force causing motion.

Lenz's Law and Back EMF

Lenz’s Law dictates that the direction of the induced emf is such that it opposes the change in the magnetic flux that produced it. This results in a back emf that counters the battery’s driving emf when the bar moves, effectively limiting the acceleration of the bar. Understanding this opposition is essential when analyzing the transient and terminal behavior of the system.

Terminal Speed

Terminal speed is the steady-state speed reached when the driving forces and the resistive forces balance each other out, resulting in zero net acceleration. In the context of this problem, it occurs when the induced motional emf exactly counters the battery’s emf, leading to a constant speed for the moving bar.

Newton's Second Law

Newton's Second Law, which states that Force equals mass times acceleration (F = ma), is fundamental to analyzing the dynamics of the system. It is used to calculate the acceleration of the bar under the influence of the net force produced by the difference between the battery voltage and the counteracting induced emf.

Transcript

00:01 Hi, in the given problem, length of the conducting bar is given as l is equal to 0 .36 meter.

00:15 Magnetic field, which is directed into the plane of paper, is given as 2 .4 tesla, that is into the page.

00:34 Emf of the battery applied to the left of the rails is given as 12 .0 volt and resistance of the circuit is 5 .0 oom.

00:52 Now there are two types of current in this rectangular loop.

00:58 One is the current due to the battery.

01:02 So battery current i b is equal to that is given simply by home's law emf by resistance and another is induced current that is i induced is equal to that is emf motional emf induced across the bar divided by resistance and that emf induced is given by the expression b vl divided by r now as per lenses law, this induced current will be opposing the battery current.

01:53 So the net current passing through this loop will be i is equal to i b minus i induced.

02:06 Or we can say this is epsilon by r minus bvl by r.

02:12 Or finally this is epsilon minus bv l by r.

02:18 Hence there will be a magnetic force acting on this current carrying bar.

02:24 So magnetic force acting on the current carrying straight bar as it has been kept in a magnetic field.

02:45 So that is given as f is equal to b i l.

02:50 Hence here it will become b into e minus bv l by r.

03:00 Into l.

03:02 And using newton's second off motion, we can say we can equate this force by the product of mass of the bar with its acceleration.

03:11 Hence, expression for acceleration produced in this bar will be a is equal to e minus b, vl, bracket closed, b, l divided by m into r.

03:33 And that is given as acceleration is given as the differentiation of velocity, means dv by d t, acceleration is time derivative of the speed.

03:43 So finally rearranging them, we get 1 by e minus bvl into dv, means integration of this 1 minus 1 upon e minus bvl with respect to dv is equal to b .l by mr dt integration of time.

04:05 So now integrating both the sides with proper limits we get integration of left end side gives us 1 by p minus bv l integration with respect to dv and having the limits from 0 to v is equal to integration of this bl divided by m r with respect to time having the limits from 0 to t so we can say integration of this 1 by e minus bvl will come out to be natural logarithm of e minus bvl then divided by the differentiation of this function so, differentiation of this bvl with respect to v.

05:07 V will be just bl with its negative sign.

05:10 And the limits from 0 to v.

05:14 Similarly here, this bl by mr, these are the constants having no integration.

05:21 And finally, the integration of dt will be just t from 0 to t.

05:25 So plugging in the limits here, it becomes minus 1 by bl first, 4.

05:35 V this is v means natural logarithm of e minus b v l then in place of b in place of v this will be so leaving behind just natural logarithm of epsilon similarly here this is b l by m r into t minus zero or we can say shifting this b l towards right hand side and multiplying the bracket with this negative sign the terms will be interchanged so this is natural logarithm of epsilon of epsilon minus b vl is equal to b square l squared by m r into t or using the rules for logarithm this is natural logarithm of e by e minus b vl is equal to b square l square by m r t or converting this logarithmic form into exponential form we get e by e minus b v l is equal to e ratio the power b square l square by mr into t or making inverse of both the it becomes e minus bvl divided by e is equal to e minus b square l square by m into r into t.

07:18 Now making a partial fraction here 1 minus bv l by e is equal to e ratio to bar minus b square l squared by m into r t or this bvl by epsilon becomes 1 minus e minus b square l square by m r t or finally expression for velocity for the speed achieved by the bar comes out to be e by b l bracket 1 minus e ratio to the bar minus b square l is square by m r now putting the given values here to find these values first of all the value of e by b will come out to be 12 .0 volt emf of the battery divided by magnetic field, which was 2 .4 tesla into length of the bar was 0 .36 meter.

08:22 Hence, this value comes out to be approximately 14 meter per second.

08:28 And then this b square, l squared by mr, it becomes for b again, this is 2 .4 to the whole square...

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