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प्रश्न
Derive the expression for the force on a current-carrying conductor in a magnetic field.
उत्तर
- The force experienced is equal to the sum of Lorentz forces on the individual charge carriers in the wire.
- Consider a small length of wire ‘dl’, area – A, Current – I.
- The free electrons drift opposite to the direction of the current.
- The relation between I & Vd is given by
I = neAVd
- In magnetic field, the force experienced is given by
`vec"F" = - "e" (vec"v"_"d" xx vec"B")`
n → number of free electrons per unit volume.
n = `"N"/"V"`
V = A dl - Lorentz force = n A dl × – `"e"(vec"v"_"d" xx vec"B")`
dF = -enA dl `(vec"v"_"d" xx vec"B")` - Current element, `"I"vec"dl" = - "enA"vec"v"_"d"`
`therefore "d"vec"F" = ("l" vec"dl" = vec"B")`
`vec"F" = ("l" vec"l" xx vec"B")`
F = BIl sin θ
Case (i):
If the conductor is along the magnetic field; θ = 0° ; F = 0.
Case (ii):
If the conductor is perpendicular to the magnetic field; θ = 90°; F = BIl.
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