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Show that the relevant force on charge –q simplifies to a Hook’s law restoring force: F = -kx, where "k" is now made up of several constants from this problem. What is k? d. Using Newton's 2nd Law, F = ma = m² d²x determine the equation of motion for the particle as dt2 it undergoes oscillations due to this Hook's law force. Determine the period T of oscillations charge –q makes about the equilibrium point z = 0 (this should not involve any calculations, see equations 15.32 and 15.37 from your book). e. Assess the validity of your solution by analyzing the physical units to make sure that your solution has units of time. → 0 and Q → o. What does the mathematical What happens to the period, T, in the limit as Q statement 0 → 0 and 0 → 0 mean physically? Does vour boss's design work? Explain in a few

Show that the relevant force on charge –q simplifies to a Hook’s law restoring force: F = -kx, where "k" is now made up of several constants from this problem. What is k? d. Using Newton's 2nd Law, F = ma = m² d²x determine the equation of motion for the particle as dt2 it undergoes oscillations due to this Hook's law force. Determine the period T of oscillations charge –q makes about the equilibrium point z = 0 (this should not involve any calculations, see equations 15.32 and 15.37 from your book). e. Assess the validity of your solution by analyzing the physical units to make sure that your solution has units of time. → 0 and Q → o. What does the mathematical What happens to the period, T, in the limit as Q statement 0 → 0 and 0 → 0 mean physically? Does vour boss's design work? Explain in a few

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PART D AND E ONLY. LAST 2 PARTS. 

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Consider a charge -q located a small distance z above the center of a positively а. charged ring with total charge +Q and radius R (see example 23.4). Define a coordinate system and draw a pictorial representation of this physical situation. Clearly label any quantities of interest on your diagram. b. Determine the net force exerted on the charge -q due to the positively charged ring. What is the magnitude of the force on the charge -q if it is at the location z = 0? Solve: Use the binomial expansion as a mathematical tool to understand the motion of the charged particle under the assumption that it is only displaced a small distance z « R away from the center of the ring. The binomial approximation to first order is written: с. (1+ x)" - 1+ nx if х «1 Using the binomial approximation, simplify your expression from part (b). You should have two terms in your expression for the net force. Hint: It may be helpful to factor the denominator such that it is in a form (1 + x)" where x is a small quantity composed of variables from the problem. Show that the relevant force on charge -q simplifies to a Hook’s law restoring force: F = -kx, where "k" is now made up of several constants from this problem. What is k? d. d²x Using Newton's 2nd Law, F = ma = m it undergoes oscillations due to this Hook's law force. Determine the period T of oscillations charge determine the equation of motion for the particle as dt2> 0 (this should not involve any calculations, -ą makes about the equilibrium point z = see equations 15.32 and 15.37 from your book). е. Assess the validity of your solution by analyzing the physical units to make sure that your solution has units of time. What happens to the period, T, in the limit as Q statement Q → 0 and Q → 00. What does the mathematical → 0 and Q → o mean physically? Does your boss's design work? Explain in a few sentences.
Transcribed Image Text:Consider a charge -q located a small distance z above the center of a positively а. charged ring with total charge +Q and radius R (see example 23.4). Define a coordinate system and draw a pictorial representation of this physical situation. Clearly label any quantities of interest on your diagram. b. Determine the net force exerted on the charge -q due to the positively charged ring. What is the magnitude of the force on the charge -q if it is at the location z = 0? Solve: Use the binomial expansion as a mathematical tool to understand the motion of the charged particle under the assumption that it is only displaced a small distance z « R away from the center of the ring. The binomial approximation to first order is written: с. (1+ x)" - 1+ nx if х «1 Using the binomial approximation, simplify your expression from part (b). You should have two terms in your expression for the net force. Hint: It may be helpful to factor the denominator such that it is in a form (1 + x)" where x is a small quantity composed of variables from the problem. Show that the relevant force on charge -q simplifies to a Hook’s law restoring force: F = -kx, where "k" is now made up of several constants from this problem. What is k? d. d²x Using Newton's 2nd Law, F = ma = m it undergoes oscillations due to this Hook's law force. Determine the period T of oscillations charge determine the equation of motion for the particle as dt2> 0 (this should not involve any calculations, -ą makes about the equilibrium point z = see equations 15.32 and 15.37 from your book). е. Assess the validity of your solution by analyzing the physical units to make sure that your solution has units of time. What happens to the period, T, in the limit as Q statement Q → 0 and Q → 00. What does the mathematical → 0 and Q → o mean physically? Does your boss's design work? Explain in a few sentences.
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