A heavy conductor (mass m, length l, resistance R) is suspended by two springs each with spring constant k, and connected to a battery with electric potential V as shown in the figure. A magnetic field B is now imposed. The acceleration of gravity is 9.8 m/s^2.
I'm going to describe this image as best as I can, as the site won't let me link to an outside image: There is a battery with a wire coming out horizontally from each side. The positive side of the battery is on the left. There is a spring hanging down from the end of each wire. Attached to the bottom of the springs is a long rod, with a spring attached to either end, allowing it to hang parallel to the ground.
What is the minimum magnetic field strength required to completely take the weight of the heavy conductor off the springs if the potential voltage is 20V, conductor resistance is 11 ohms, length is .63 m, spring constant is 1.7 N/m, conductor mass is 1.6 kg?
F(spring) = -kx
F(b) = I*L x B (cross product)
where L is the vector in the direction of I w/ magnitude of the conductor length l and F(b) is the resultant force exerted by a magnetic field on a current-carrying wire (I assume this can also be applied to a current-carrying conductor)
The Attempt at a Solution
I drew a free body diagram for the conductor. The force downward is mg. The forces upward and opposite mg are 2kx (that is: F(net spring) = k1*x + k2*x, and k1 and k2 are equal), and I*L x B.
This is my reasoning: in order for the weight to be completely taken off of the springs, the upward force F(b) must equal the downward force, which is mg. Therefore, the spring force does not need to be taken into consideration.
mg = I*L x B
mg = (I * L-sub-x * i-hat) x (B-sub-z * k-hat) <--since the only value of L is in the y-direction, and the only value of B is in the z-direction
mg = -I * L-sub-x * B-sub-z * j-hat
B-sub-z * j-hat = -mg/(I*L-sub-x)
Plugging in the numbers gives me -45.252525...
This answer is incorrect.
Am I reasoning the FBD correctly? Am I right that I should just set F(b) equal to mg?