Answer to Question #129320 in Electric Circuits for Faith

Question

A metal crossbar of mass m slides without friction on two parallel conducting rails a distance b apart. A a resistor R is conducted across the rails at one end. Compared with R the resistance of the bar is negligible. There is a uniform field perpendicular to the plane of the figure. Does the rod ever stop moving ? And if so when?

Answer 1

The answer is "it will stop moving only after infinite period of time".

When the rod moves in magnetic field, it experiences the Ampere's force $F_A$. The current direction can be found using Fleming's right-hand rule and is shown on the picture. Magnetic force direction can be found using Fleming's right-hand rule and is also shown. If the rod moves in another direction, the current direction changes to opposite as well force direction.Thus the force is always opposite to the velocity hereby decreasing it.

According to Faraday's law of induction

$ε=-dΦ/dt=-BdS/dt=-B*b*dl/dt.$

According to Ohm's law

$ε=IR$.

Hence,

$IR=-B*b*dl/dt \Rightarrow I=-(Bb)/R*dl/dt=-Bbv/R$. (1)

On the other hand, Ampere's force:

$F_A=BIb.$

From the second Newton`s law of motion

$F_A=mdv/dt.$

Hence,

$mdv/dt=BIb \Rightarrow I=m/(Bb)*dv/dt$ . (2)

From the equations (1) and (2):

$m/(Bb)*dv/dt=-Bbv/R \Rightarrow \intop_{v_0}^0dv/v=$

$=-B^2b^2/(Rm)*\intop_0^tdt;$

$ln(0)=-B^2b^2t/(Rm)$ .

As we can see, the time is infinite, so the rod won`t stop. It can be explained in this way:

When the rod moves in magnetic field its mechanical energy converts into elecricity, which is dissipated on the resistor. The more energy dissipates, the less velocity has the rod, the less is the intensity of energy dissipation. According to this it will never stop as well.