Explaining F=BIl and F=Bqv for Many Charges

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F = BIl and F = Bqv are equivalent expressions of the Lorentz Force Law, representing the force on a current-carrying wire and a moving charge, respectively. In the case of a wire of length l carrying a net charge q with average velocity v, the current I can be expressed as I = q/t, leading to the relationship F = BIl. The equation F = Bqv applies to a single charge moving with velocity v in a magnetic field B. Both equations involve the cross product, indicating that the force, current, and magnetic field vectors are mutually orthogonal. This demonstrates how the principles governing the force on a wire and a moving charge are fundamentally linked.
poojarao
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Speed = distance/time and current = charge/time. Explain how F=BIl is actually the same equation as F= Bqv but considered for many charges in a group?

can someone please explain with working out please?

thanks
 
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Lets say that you had a wire with a length l. Let's say that traveling through that length is a net charge q, with an average velocity of v. v = distance/time, in which in this case, v = l/t. The current is I = charge/time = q/t. Therefore, F = B*q*(l/t) = B*(q/t)*l = BIl.
 
thanks a lot!
 
For clarity, these two equations (part of the Lorentz Force Law) are written as

F = l(I x B) = q(v x B)

where x indicates the cross product. meaning that the vectors F,I and B; or F,v,and B are all mutually orthogonal.
 
The I/v vectors need not be orthogonal with the B vector.
 
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