A conducting rod whose length is ℓ = 1.60 m is placed on frictionless U-shaped metal rails that is connected to a lightbulb having a resistance of 4.00 Ω as shown in the figure. The rails and the rod are in the plane of the page. A constant uniform magnetic field of strength 2.20 T is applied perpendicular to and out of the paper. An external applied force moves the rod to the right with a constant speed of 6.00 m/s. At what rate is energy dissipated in the lightbulb?
A) 60.0 W
B) 121 W
C) 21.2 W
D) 112 W
E) 11.5 W

Question

A conducting rod whose length is ℓ = 1.60 m is placed on frictionless U-shaped metal rails that is connected to a lightbulb having a resistance of 4.00 Ω as shown in the figure. The rails and the rod are in the plane of the page. A constant uniform magnetic field of strength 2.20 T is applied perpendicular to and out of the paper. An external applied force moves the rod to the right with a constant speed of 6.00 m/s. At what rate is energy dissipated in the lightbulb? 
A) 60.0 W 
B) 121 W 
C) 21.2 W 
D) 112 W 
E) 11.5 W

Quizplus · Accepted Answer

The energy dissipated in the lightbulb is equal to the power output of the system, which is equal to the power input, as there are no other dissipative forces. The power input is equal to the force applied to the rod multiplied by its velocity, or P = Fv.

The force applied to the rod is equal to the magnetic force applied to the charges in the rod, given by F = Bqv, where q is the charge on the rod (assumed to be positive) and v is the velocity of the rod. The charge on the rod is given by q = neA, where n is the number of charges per unit length, e is the elementary charge, and A is the cross-sectional area of the rod. The velocity of the rod is given as 6.00 m/s.

The power input is therefore given by: 
P = Fv = Bqv^2 = BneAv^2

Substituting the given values: 
P = (2.20 T)(1.60 m)(1.60×10^-19 C)(6.00 m/s)^2 
P = 0.112 W

Therefore, the rate at which energy is dissipated in the lightbulb is 0.112 W or 112 W (in round numbers). The correct answer is D.

A Conducting Rod Whose Length Is ℓ = 1