Average force on an electron in a wire

Hey, just did a 25 question multiple choice paper and this was the only one that I got wrong. Was hoping someone could enlighten me.

Quote:

Originally Posted by **AQA PHYA4/1 Jan 2010**

A current of

is passed through a conductor of length

and cross-sectional area

. The conductor contains

electrons per

. When the conductor is at right angles to a magnetic field of flux density

, it experiences a magnetic force. What is the average magnetic force that acts on one of the free electrons in the wire?

I got the force acting on the conductor:

So figured since that is the force on the entire conductor, then the force on one electron would be the force divided by the number of electrons:

But the answer is

I know I am neglecting the cross-sectional area but not sure where it would come in.

Thanks.