A thin uniform rod with negligible mass and length ℓ is attached to the floor by a frictionless hinge at point P. A horizontal spring with force constant k connects the other end to wall. The rod is in a uniform magnetic field B directed into the plane of paper what is extension in spring in equilibrium when a current is passed through the rod in direction shown. Assuming spring to be in natural length initially.

Engineering

Physics

Equilibrium

SHM of Spring Block System

Magnetic Force on a Current Carrying Wire

Question

A thin uniform rod with negligible mass and length ℓ is attached to the floor by a frictionless hinge at point P. A horizontal spring with force constant k connects the other end to wall. The rod is in a uniform magnetic field B directed into the plane of paper what is extension in spring in equilibrium when a current is passed through the rod in direction shown. Assuming spring to be in natural length initially.

$\frac{3 iℓB}{8 k}$

$\frac{5 iℓB}{8 k}$

$\frac{5 iℓB}{6 k}$

$\frac{5 iℓB}{4 k}$

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Torque due to magnetic force

$| d \vec{𝛕} | = \int_{0}^{L} (idrB) r$

= $\frac{{iL}^{2} B}{2}$

In equilibrium

$\frac{{iL}^{2} B}{2} = (kx) \times Lsin 30^{\circ}$

or $x = \frac{5 iLB}{8 k}$