An air-filled capacitor is formed from two long conducting cylindrical shells that are coaxial and have radii of 48 mm and 84 mm. The electric potential of the inner conductor with respect to the outer conductor is -400 V. (k = 1/4πε₀ = 8.99 × 10⁹N ∙ m²/C²) The energy stored in a 1.0-m length of this capacitor is closest to A) 8.0 μJ. B) 5.7 μJ. C) 11 μJ. D) 16 μJ. E) 22 μJ.

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Quizplus · Accepted Answer

The energy stored in a capacitor is given by:$$U = \frac{1}{2}CV^2$$where C is the capacitance and V is the potential difference. The capacitance of a cylindrical capacitor is given by:$$C = 2\pi\epsilon_0\frac{L}{ln(b/a)}$$where L is the length of the capacitor, a is the radius of the inner conductor, and b is the radius of the outer conductor. Substituting these expressions into the equation for energy, we get:$$U = \frac{1}{2}\left(2\pi\epsilon_0\frac{L}{ln(b/a)}ight)V^2$$Plugging in the given values, we get:$$U = \frac{1}{2}\left(2\pi(8.99 	imes 10^9 	ext{ N}\cdot	ext{m}^2/	ext{C}^2)\frac{1.0 	ext{ m}}{\ln(84 	ext{ mm}/48 	ext{ mm})}ight)(-400 	ext{ V})^2$$$$U = 8.0 \mu	ext{J}$$

An Air-Filled Capacitor Is Formed from Two Long Conducting Cylindrical