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Transcript

Homework Problem Set 8 Solutions 1) How is an LRC circuit similar to a mechanical harmonic oscillator? Which component of the circuit acts like a “mass”? Which component of the circuit acts like “friction”? An LRC circuit is similar to a mechanical oscillator because they alternate between potential and kinetic energy. The component of the circuit that acts like a mass is the inductor and the component that acts like friction is the resistor. 2) Use Gauss’ Law and Ampere’s Law to find both the capacitance per unit length and the inductance per unit length of a coaxial cable with an outer radius of 4.5 mm and an inner radius of 1.5 mm. Assume the space between the two conductors is filled with air. Solving for capacitance per unit length: ⃗ ∙ 𝒅𝑨 ⃗ ∯𝑬 𝒒 𝜺𝟎 ∯|𝑬||𝒅𝑨| = |𝑬| = 𝒓𝟐 𝒒 𝜺𝟎 ∯ 𝒅𝑨 𝒓𝟐 ⃗ ∙ 𝒅𝒔 ⃗ ∙ 𝒅𝒓 ⃗ =∫ 𝑬 ⃗ = ∫ |𝑬|𝒅𝒓 = ∫ 𝑽 = ∫𝑬 𝒓𝟏 𝒓𝟏 𝑪= 𝒓𝟐 𝒓𝟏 𝑸 = 𝒓 = 𝒒 𝒒 = 𝜺𝟎 𝜺𝟎 𝟐𝝅𝒓𝒍 𝒓𝟐 𝒒𝒅𝒓 𝒒 𝒅𝒓 𝒒 𝒓𝟐 = ∫ = (𝒍𝒏 ( )) 𝜺𝟎 𝟐𝝅𝒓𝒍 𝜺𝟎 𝟐𝝅𝒍 𝒓𝟏 𝒓 𝜺𝟎 𝟐𝝅𝒍 𝒓𝟏 𝒒 𝜺𝟎 𝟐𝝅𝒓𝒍 𝒒 𝒓𝟐 = 𝒓 𝒍𝒏 (𝒓 ) 𝒍𝒏 (𝒓𝟐 ) 𝜺𝟎 𝟐𝝅𝒓𝒍 𝟏 𝟏 𝟐𝝅 (𝟖. 𝟖𝟓 × 𝟏𝟎−𝟏𝟐 𝑪𝟐 ) 𝑵 ∙ 𝒎𝟐 𝑪 𝜺𝟎 𝟐𝝅𝒓 = = 𝒍 𝒍𝒏 (𝒓𝟐 ) 𝒍𝒏(𝟑) 𝒓𝟏 Solving for Inductance per unit length: = 𝟓𝟎. 𝟔 𝒑𝑭 𝑭 𝒐𝒓 𝟓. 𝟎𝟔 × 𝟏𝟎−𝟏𝟏 𝒎 𝒎 ⃗ ∙ 𝒅𝒔 ⃗ = 𝝁𝟎 𝑰 ∮ ⃗𝑩 ∮|𝑩||𝒅𝒔| = 𝝁𝟎 𝑰 |𝑩| = 𝝁𝟎 𝑰 𝟐𝝅𝒓 𝜇0 𝐼 ⃗ ∙ 𝒅𝑨 ⃗ 𝐿 Ф𝐵 ∯ 𝑬 ∯|𝑬||𝒅𝑨| ∯|𝑩|𝒍𝒅𝒓 ∯|𝑩|𝒅𝒓 ∯ 𝟐𝝅𝒓 𝒅𝒓 = = = = = = 𝑙 𝐼𝑙 𝐼𝑙 𝐼𝑙 𝐼𝑙 𝐼 𝐼 −7 𝑇 4𝜋 × 10 𝜇0 𝒅𝒓 𝜇0 𝒓𝟐 = ∫ = 𝒍𝒏 ( ) = 𝟐𝝅 𝒓 𝟐𝝅 𝒓𝟏 𝟐𝝅 ∙𝑚 𝐴 𝒍𝒏(𝟑) = 𝟐. 𝟐 × 𝟏𝟎−𝟕 𝑯 𝒎 3) An AC generator with a variable frequency is connected across a 45 cm piece of the coaxial cable described in Question 2. If the cable has minimal resistance, at what frequency will the cable resonate? If the resistance of the cable is 2Ω per meter, what is the new resonant frequency? a) 𝜔0 = 1 √𝐿𝐶 = 1 √𝑙 2 𝐿𝐶 = 1 𝑯 𝑭 √(.45)2 (𝟐.𝟐×𝟏𝟎−𝟕 )(𝟓.𝟎𝟔 ×𝟏𝟎−𝟏𝟏 ) 𝒎 = 670𝑀𝐻𝑧 𝒎 b) The new resonant frequency is the same as the 𝜔0 , 1.4 × 10-7 Hz, because resonance frequency is not a function of resistance and therefore has no effect on the frequency. 4) Two circular loops of wire with radius 8.5 cm are sitting one on top of another such that the total flux through the first loop travels through the second loop, and vice versa. What is their mutual inductance? In this case the mutual inductance of the two loops is the same as the self-inductance of a single circular loop of wire. For a loop of radius ‘r’ and wire radius ‘a’ this inductance is given by: 8𝑟 ) − 2) 𝑎 𝑇∙𝑚 8(.085𝑚) 𝐿 = (4𝜋 × 10−7 ) (.085)(ln ( ) − 2) 𝐴 𝑎 .68 𝐿 = 1.1 × 10−7 (ln ( 𝑎 ) − 2) H 𝐿 = 𝜇0 𝑟(ln (