Question

Two parallel metal rods form a plane inclined at 20.0° to the horizontal. The upper end of the rods are connected together by a resistor of 122.0 Ω. The lower end of the rods are held in an electrically insulating block. They are separated horizontally by a distance of 20.0 cm There is a uniform magnetic field of 0.100 T directed vertically upward. A metal bar of mass 0.10 g is sliding down the metal rods. What is the maximum speed of the bar? Assume: no frictional loss and that the rods are very long; g=9.8 m/s2.

Answer 1

Hi,

In this case, we have a bar that is falling down over two metal rods. The bar falls due to the action of gravity. However, as the bar falls, the two rods and the resistence form an electric circuit. As we have a conductor in movement while there's a magnetic field applied over it, a magnetic force that oppose the movement of the bar applies following Faraday Law of Induction.

If we think about the movement of the bar, once it begins its movement, the speed is cero. Then the gravity and the magnetic force compete with each other, and at the first moments, the gravity is bigger, so it appears a positive acceleration and the speed increases its value. However, as time passes by, the magnetic force grows to a point where it is equal to the force produced by gravity and after that time the bar begins to slow down.

Therefore, the moment where the speed of the bar is maximum when the weight of it and the magnetic force are equal.

Now we only need the appropriate equations to solve this problem:

I = (1/R) BLvcos(θ) (1); where I is the current (in this case induced), L is the lenght of the bar in movement, v is the speed of the bar, R is the resistance of the circuit, B is the magnetic field and θ is the angle between the vertical axis and the magnetic field.

F = ILB sin (θ) (2); where F is the magnetic force, I is the current, B is the magnetic field and θ is the angle between the current and the magnetic field.

F = mg (3) ; where m is the mass of the bar and g is the acceleration due to gravity.

In this case we use (3) to find the value of the force, then we use (2) in order to find the current and then we use (1) to find the value of the speed (maximum, in this case).

F = (0.1*10^-3 kg)*(9.8 m/s²) = 9.8*10^-4 N

I = F / (LB sinθ) = 9.8*10^-4 N / [ (0.2 m)*(0.1 T)*(sin(70°)) ] = 0.052 A

v = IR/(LB cosθ) = (0.052 A)*(122 Ω) / [(0.2 m)*(0.1 T)*cos(70°)] = 927.4 m/s

The speed is quite big, so it is logic the supposition of very long rods.

I hope it helps.