# Net Lorentz force?

From the following diagram: Each (golden) wire is exposed to the magnetic field $B$, while as the black wires are not. Assume current supplied from the power supply to be $50A$, length $L$, then the Lorentz force formula is:

$$F = IL \times B$$

When $F$ is calculated, to find the total force ,is it multiplied $F$ by 3? the $F_t$ = $3(F)$

If so, how can that be true when current is $50A$? A bit confused how at the same current we could magnify force, of course at the cost of higher power due to increase of resistance?

• As drawn, $F$ is the force on each (golden) wire. What do you mean by the total force, i.e., total force on what? – Alfred Centauri Jan 29 '15 at 18:16
• Total Lorentz force, on the system. IF for example, they are connected to one another. – Pupil Jan 29 '15 at 19:14
• Force does not equal to power. Force can be sustained without using any power. For example, if the wire has zero resistance (superconductor) and fixed in place. The 50A current flow uses no power whether it passes over the magnetic field, one or three or a thousand times. This is an important ingredient for maglev for example. Another analogy, stick a steel ball under a permanent magnet, no power is used by the magnet to keep the steel ball hanging. Change to a magnet three times the strength, the steel ball sticks to the magnet with three times the force, but still no power. – rioraxe Jan 30 '15 at 5:47

If each 'golden' wire has a length L, then you've multiplied L by 3 when you have 3 of those wires each exposed to the field.
3 times the wire length will require 3 times as much voltage from PS in order to maintain 50A.
So with 3 times the voltage and the same current you're delivering 3 times the power.

• But is it true, that the force is 3 times more? If the golden wires are mechanically connected to each other? – Pupil Jan 31 '15 at 6:54

An alternative explanation:

The force f on a single charge is:

$$f= B.q.v$$

If you assume that the charges are distributed uniformly over the length of a wire, there are simply 3x more charges experiencing this force. It's as if 3 times more people are pulling the same trolley.

Every charge in the conductor is subject to the same force. This is where F= B.I.l actually comes from.

Force on a single charge:

$$f= B.q.v$$

or for multiple charges:

$$F= B.(n.q).v = B.Q.v$$

Current is just rate of change of charge in time. If charge is distributed uniformly over the length of the wire:

$$I = \frac{Q}{t}$$

$$F = B.I.t.v = B.I.l$$

• Well it make sense, looking back at the diagram and at brhan's answer it shows clearly why the force is 3times more, the length too. The total length is 3 times if we consider all there golden wires. We could simply use L to be Lx3 too. – Pupil Jan 31 '15 at 7:03

If the wire doesn't move, your are right, then only the resistance is limiting the current in your wire. This is comparable to what is happening if a motor is blocked.

As soon as the wire starts to move, the magnetic field is inducing a voltage (called back EMF), such that it is reducing the current. The induced voltage minus the external voltage times the current (power) is then equal to force times speed (again power). The law of conservation of energy isn't violated.

The induced voltage is proportional to L. In case of a motor you are trading force for speed (but keep power constant), by adding more windings.