A uniform non-conducting disc of radius r and mass m is charged uniformly with the charge q. This disc is placed flat on a rough horizontal surface having coefficient of friction µ. A uniform magnetic field is present in a circular region (a > r > 0) but varying as kt3 as shown in figure. Find the time in second after which the disc begins to rotate. (Given r = 1 m, m = 18 kg, q = 1C, µ = 0.1, K = 4, g = 10 m/s2)

Engineering

Physics

Magnetic Flux and Faraday Law and Lenz Law

Static and Kinetic Friction

Question

A uniform non-conducting disc of radius r and mass m is charged uniformly with the charge q. This disc is placed flat on a rough horizontal surface having coefficient of friction µ. A uniform magnetic field is present in a circular region (a > r > 0) but varying as kt³ as shown in figure. Find the time in second after which the disc begins to rotate. (Given r = 1 m, m = 18 kg, q = 1C, µ = 0.1, K = 4, g = 10 m/s²)

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$E = \frac{x}{2} \frac{dB}{dt}$

$E = \frac{3 {Kxt}^{2}}{2}$

$d τ = \frac{3 {Kxt}^{2}}{2} \times \frac{2 πxdx}{{πr}^{2}} q . x .$

$τ = \frac{3 {Kt}^{2} q}{r^{2}} \int_{0}^{r} x^{3} dx$

$τ = \frac{3 Kq . t^{2}}{4} . r^{2}$ .......(i)

torque due to friction force

dτ = µdmgx

$τ = 2 μg \frac{qm}{r^{2}} \int_{0}^{r} x^{2} dx = \frac{2}{3} μmgr$ ....(ii)

$\frac{3 Kq . t^{2} r^{2}}{4} = \frac{2}{3} = µ mgr$

$t = \sqrt{\frac{8 µ mg}{9 Kqr}} = 2 seconds .$