Name: Conversion of a Galvanometer Into a Voltmeter
Uploaded: 2019-07-19
Duration: 7 min 18 s
Description: This video helps us understand the working of a galvanometer. It also teaches us how to convert it into a voltmeter of the desired range both theoretically a...

Question

Find the force acting on a current carrying conductor in uniform magnetic field. Using it find the force between two parallel current carrying conductors.

Accepted Answer

Consider a straight conductor $P Q$ of length $l$ placed along y-axis and an uniform magnetic field $\vec{B}$ along z-axis i.e. perpendicular to the length of conductor.

If current $I$ flows through the conductor $P Q$ such that it is parallel along the conductor, current flows due to drifting of electron.

If $n$ : number of free electrons per unit volume, $A$ : cross-sectional area of the conductor and $v_{d}$ : drift velocity

Then, current flowing through the conductor is: $I = n e A v_{d}$ ........................ $(1)$

Multiplying both sides by $l$ , we get: $I l = - n e A l v_{d}$ ........................ $(2)$

Here, negative sign is due to flow of electron is opposite to current direction.

Now, the magnetic force on the electron: $B_{e} = - e (\vec{v_{d}} \times \vec{B})$ ........................ $(3)$

If $n$ is total numbers of free electrons in the conductor then force on the current carrying conductor placed inside the magnetic field is: $F = N B_{e}$ ........................ $(4)$

Substituting, $N = n A l$ and equation $(3)$ in equation $(4)$ , we get, $F = n A l [- e (\vec{v_{d}} \times \vec{B})]$ ........................ $(5)$

$\Rightarrow F = (- n e A l \vec{v_{d}}) \times \vec{B}$

$\Rightarrow F = I \vec{l} \times \vec{B} \Rightarrow F = B I l \sin θ$ ........................ $(6)$

Hence, the force acting on a current carrying conductor in uniform magnetic field will be $B I l \sin θ$ .

Force between two parallel current carrying conductors:

Consider two infinitely long conductors placed parallel to each other at a distance $r$ apart.

Magnetic field at point on a conductor $1$ due to conductor $2$ : $B_{2} = \frac{μ_{\circ}}{2 π} \frac{I_{2}}{r}$ ........................ $(7)$

Force experienced by conductor $1$ per unit length due to $B_{2}$ : $F = I_{1} \vec{d l} \times \vec{B_{2}}$ ........................ $(8)$

Here, $θ = 90 °$ , substituting equation $(7)$ in equation $(8)$

$F = \frac{μ_{\circ}}{2 π} \frac{I_{1} I_{2}}{r}$ ........................ $(9)$

Hence, the force between two parallel current carrying conductors is $\frac{μ_{\circ}}{2 π} \frac{I_{1} I_{2}}{r}$ .

Find the force acting on a current carrying conductor in uniform magnetic field. Using it find the force between two parallel current carrying conductors.

Important Questions on Moving Charges and Magnetism

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