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A hollow, right circular cone of base radius R and height h, with its tip at the origin is rotating about the Z-axis with an angular velocity ω, as shown in the figure. The cone carries a total charge Q uniformly distrib

Concept: Magnetic field of rotating charge

Question

A hollow, right circular cone of base radius R and height h, with its tip at the origin is rotating about the Z-axis with an angular velocity ω, as shown in the figure. The cone carries a total charge Q uniformly distributed on its curved surface. The magnitude of magnetic field at a point (0,0, z), where z >> R and z >> h, is nμ0QR2ω4πz3\frac{n\mu_0 QR^2\omega}{4\pi z^3}. The value of n is:
B=μ0IR22(R2+z2)3/2B = \frac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}}

Try it yourself first

Hint: Think about how the charge distribution on the cone's surface relates to the current created by its rotation. How would you express a small element of charge dQ on a ring of the cone?

Step-by-step solution

  1. 1

    Differential current

    Differential ring of charge • Rotating charge → current • dI = (dQ/2π)ω
    dI=dQ2πωdI = \frac{dQ}{2\pi}\omega
  2. 2

    Magnetic field of a ring

    Magnetic field of a ring • dB = (μ₀ dI r²)/(2(r²+z')^(3/2)) • z' is distance from ring to point
    dB=μ0dIr22(r2+z′2)3/2dB = \frac{\mu_0 dI r^2}{2(r^2+z'^2)^{3/2}}
  3. 3

    Integrate over cone

    Integrate dB • r = (R/h)z' • dQ = (Q/Area)dA
    B=∫dBB = \int dB
  4. 4

    Far field approximation

    z >> R, z >> h • Denominator (r²+z'²)^(3/2) ≈ z³ • B ≈ (μ₀ Q R² ω)/(4π z³)
    B≈μ0QR2ω4πz3B \approx \frac{\mu_0 Q R^2 \omega}{4\pi z^3}
  5. 5

    Find n

    Compare with given formula • n = 2

Final answer

2

Remember: Magnetic field of a rotating charged body can be found by integration.

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If a charged disk rotates about its central axis, which of the following statements about the magnetic field at its center is true?

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