A conducting wire of parabolic shape, initially is moving with velocity in a non-uniform magnetic field as shown in the figure. If , and are positive constants and is the potential difference developed between the ends of the wire, then the correct

Question

A conducting wire of parabolic shape, initially $y = x^{2}$ is moving with velocity $\vec{V} = V_{0} \hat{i}$ in a non-uniform magnetic field $\vec{B} = B_{0} (1 + {(\frac{y}{L})}^{β}) \hat{k},$ as shown in the figure. If $V_{0} B_{0}$ , $L$ and $β$ are positive constants and $Δ ϕ$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are: [2019]

Smart Achievers · Accepted Answer

(1, 2, 4)

Given: $B = B_{0} [1 + {(\frac{y}{L})}^{β}], \vec{V} = V_{0} \hat{i}$

We now consider an infinitesimally small length of wire $d y$ at a distance $y$ from the origin.

Emf induced across the length dy $| ∆ ϕ |$ $= B (d y) V_{0}$

$| ∆ ϕ |$ $= B_{0} [1 + {(\frac{y}{L})}^{β}] V_{0} d y$

$∴$ Induced emf across the complete projection

$| Δ ϕ |$ $= B_{0} V_{0} \int_{0}^{L} [1 + {(\frac{y}{L})}^{β}] V_{0} d y$ $= B_{0} V_{0} L [1 + \frac{1}{β + 1}]$

For $β = 0$ , $| Δ ϕ |$ $= 2 B_{0} V_{0} L$

Clearly, $| Δ ϕ |$ $\propto L$

For $β = 2$ , $| Δ ϕ |$ $= \frac{4}{3} B_{0} V_{0} L$

For a straight wire of length $\sqrt{2} L$ , placed along $y = x$ , the value of $| Δ ϕ |$ will remain the same as its projection on the $y$ -axis is the same $L$ as that of the previous.

Solution

1 $| Δ ϕ |$ is proportional to the length of the wire projected on the $y$ -axis.

2 $| Δ ϕ |$ remains the same if the parabolic wire is replaced by a straight wire, $y = x$ initially, of length $\sqrt{2} L$ .

3 $| Δ ϕ |$ $= \frac{1}{2} B_{0} V_{0} L for β = 0$

4 $| Δ ϕ |$ $= \frac{4}{3} B_{0} V_{0} L for β = 2$

Want Us to Email you About Special Offers & Updates?

Solution

1 |Δϕ| is proportional to the length of the wire projected on the y-axis.

2 |Δϕ| remains the same if the parabolic wire is replaced by a straight wire, y=x initially, of length 2L.

3 |Δϕ|=12B0V0L for β=0

4 |Δϕ|=43B0V0L for β=2

Want Us to Email you About Special Offers & Updates?

1 $| Δ ϕ |$ is proportional to the length of the wire projected on the $y$ -axis.

2 $| Δ ϕ |$ remains the same if the parabolic wire is replaced by a straight wire, $y = x$ initially, of length $\sqrt{2} L$ .

3 $| Δ ϕ |$ $= \frac{1}{2} B_{0} V_{0} L for β = 0$

4 $| Δ ϕ |$ $= \frac{4}{3} B_{0} V_{0} L for β = 2$