A conducting ring of circular cross-section with inner and outer radii a and b is made out of a material of resistivity ρ . The thickness of the ring is h . It is placed coaxially in a vertical cylind

Question

A conducting ring of circular cross-section with inner and outer radii $a$ and $b$ is made out of a material of resistivity $ρ$ . The thickness of the ring is $h$ . It is placed coaxially in a vertical cylindrical region of a magnetic field $B = krt$ , where $k$ is a positive constant, $r$ is the distance from the axis and $t$ is the time. If the current through the ring is $I = (\frac{kh}{αp}) [b^{3} - a^{3}]$ , then what is the value of $α$ ?

TestHub · Accepted Answer

The flux through the area of radius r is ϕ = ∫ 0 r k r t 2 π r d r = 2 π k t r 3 3 E.M.F. induced, ϵ = dϕ dt = 2 πkr 3 3 Net induced current is I = ∫ a b 2 πr 3 khdr 3 ρ 2 πr = kh 9 ρ b 3 - a 3 ∴ α = 9

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