24- capacitors: capacitance, combination (equivalent capacity), dielectric materials, Tomorrow – Quiz 2: Thursday/Friday (Oct 9 & 10) – Gauss, electric potential, Capacitors More-or-less parallel capacitor • The membrane of the axon of a nerve cell is a thin cylindrical shell of radius r = 10-5 m, length L = 0.48 m, and thickness d = 10-8 m. The membrane has a positive charge on one side and a negative charge on the other, and acts as a parallel-plate capacitor of area A = 2πrL and separation d. Its dielectric constant is about κ = 5. Cylindrical capacitor • Gauss’s surface: cylinder of radius, r, and length L. • 𝐸 ∙ 𝑑𝑎 = cylinder above 𝑄inc 𝜀0 • →2𝜋𝑟𝐿𝐸 = 𝜆𝐿/𝜀0 →𝐸 = 𝜆/2𝜋𝑟𝜀0 • Field is inversely prop to r, not r2. • Now elec. potential difference between r = a and b! • Δ𝑉 = − 𝑎 𝐸 𝑏 ∙ 𝑑𝑟 = 𝑏 𝑑𝑟 𝜆 2𝜋𝜀0 𝑎 𝑟 = 𝜆 𝑏 ln . 2𝜋𝜀0 𝑎 • If the capacitor is thin (𝑏 = 𝑎 + 𝛿) 𝐴𝜀0 ? 𝛿 𝜆 𝑎+𝛿 𝜆 ln = ln 2𝜋𝜀0 𝑎 2𝜋𝜀0 • Can we show that 𝐶 = • 𝜆 𝑏 ln 2𝜋𝜀0 𝑎 • 𝐶= 𝑄 Δ𝑉 = = 2𝜋𝜀0 𝐿𝑎 𝛿 = 𝐴𝜀0 ! 𝛿 1+ 𝛿 𝑎 ~ 𝜆 𝛿 2𝜋𝜀0 𝑎 = 𝑄/𝐿 𝛿 . 2𝜋𝜀0 𝑎 More or less parallel capacitor • The membrane of the axon of a nerve cell is a thin cylindrical shell of radius r = 10-5 m, length L = 0.48 m, and thickness d = 10-8 m. The membrane has a positive charge on one side and a negative charge on the other, and acts as a parallel-plate capacitor of area A = 2πrL and separation d. Its dielectric constant is about κ = 5. A parallel plate capacitor is hooked up across a battery’s terminals, charging it up to a given potential. While the capacitor remains connected to the battery, the distance between the capacitor’s plates is doubled. What is the change of the energy stored in this new capacitor as compared with the original capacitor? A. Decrease by factor of 4 B. Decrease by factor of 2 C. Stays the same D. Increases by factor of 2 E. Increase by factor of 4 How about this? C1 C4 C3 C2 C5 How about this? C1 C4 C3 C2 C5 • • • • • • 𝑉1 + 𝑉4 = 𝑉 𝑉2 + 𝑉5 = 𝑉 𝑉1 − 𝑉2 = 𝑉3 𝑄1 + 𝑄3 − 𝑄4 = 0 𝑄2 − 𝑄3 − 𝑄5 = 0 5 unknowns, 5 equations? Geiger tube • E-field? • Equipotential surfaces? • Where is 500 V? • Argon ion speed as it hits the cathode? Dielectric materials +Q + + + + + + + + + + + –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –Q –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ –+ – – – – – – – – – – – +Q = + + + + + + + + + + + • Δ𝑉 = 𝐸𝑑 = 𝐸𝑄 + 𝐸𝑄′ 𝑑 = 𝑄−𝑄′ 𝑑 𝐴𝜀 = 𝑄−𝑄𝛼 𝑑 𝐴𝜀 →𝐶= –Q’ +Q’ – – – – – – – – – – – + + + + + + + + + + + 𝜎𝑄 +𝜎𝑄′ 𝑑 𝐴𝜀0 𝑑 1−𝛼 𝜀0 = = 𝜅𝐴𝜀0 𝑑 – – – – – – – – – – – –Q