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A 20.0-\(\mu\)F capacitor is charged to a potential difference of 800 V. The terminals of the charged capacitor are then connected to those of an uncharged 10.0-\(\mu\)F capacitor. Compute (a) the original charge of the system, (b) the final potential difference across each capacitor, (c) the final energy of the system, and (d) the decrease in energy when the capacitors are connected.

Short Answer

Expert verified
Original charge is 0.016 C. Final voltage is 533.33 V. Final energy is 4.2667 J. Decrease in energy is 2.1333 J.

Step by step solution

01

Calculate the Original Charge

The original charge on the 20.0-\(\mu\)F capacitor is calculated using the formula \(Q = C \times V\). Here, \(C = 20.0\,\mu\text{F}\) and \(V = 800\,\text{V}\). Therefore, \(Q = 20.0 \times 10^{-6}\,\text{F} \times 800\,\text{V} = 0.016\,\text{C}\).
02

Calculate the Final Potential Difference

The total charge is conserved, so the charge \(Q = 0.016\,\text{C}\) is shared between the two capacitors connected in parallel. The equivalent capacitance \(C_{eq}\) is the sum of the two capacitances: \(C_{eq} = 20.0\,\mu\text{F} + 10.0\,\mu\text{F} = 30.0\,\mu\text{F}\). The final voltage \(V_f\) across both capacitors is given by \(V_f = \frac{Q}{C_{eq}} = \frac{0.016\,\text{C}}{30.0 \times 10^{-6}\,\text{F}} = 533.33\,\text{V}\).
03

Calculate the Final Energy of the System

The final energy stored in the system is given by \(E_f = \frac{1}{2} C_{eq} V_f^2\). Substitute \(C_{eq} = 30.0 \times 10^{-6}\,\text{F}\) and \(V_f = 533.33\,\text{V}\) to get \(E_f = \frac{1}{2} \times 30.0 \times 10^{-6}\,\text{F} \times (533.33)^2\,\text{V}^2 = 4.2667\,\text{J}\).
04

Calculate the Original Energy

The original energy stored in the 20.0-\(\mu\)F capacitor is \(E_i = \frac{1}{2} C V^2 = \frac{1}{2} \times 20.0 \times 10^{-6}\,\text{F} \times (800)^2\,\text{V}^2 = 6.4\,\text{J}\).
05

Calculate the Decrease in Energy

The decrease in energy when the capacitors are connected is the difference between the original energy and the final energy: \(E_{ ext{decrease}} = E_i - E_f = 6.4\,\text{J} - 4.2667\,\text{J} = 2.1333\,\text{J}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrostatics
Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static electric charges. In this context, we talk about capacitors, devices used to store electrical energy in an electric field. Capacitors have two conductors separated by an insulator, and when connected to a voltage source, such as a battery, they become charged.

When you consider a charged capacitor, like the initial 20.0-\(\mu\text{F}\) capacitor charged to 800 V, it holds a charge calculated by the formula \(Q = C \times V\), where \(C\) is the capacitance and \(V\) the voltage. A charged capacitor has a potential difference across its plates and an electric field inside that exerts forces on any charge placed within the field.

In our problem, this charged capacitor is connected to an uncharged 10.0-\(\mu\text{F}\) capacitor. When the connection is made, charge redistribution occurs due to electrostatic forces, causing a new uniform potential difference across both capacitors.
Energy Conservation
Energy conservation is a fundamental principle in physics that states energy cannot be created or destroyed but can only change forms. In electrostatic systems, energy is conserved during charge redistribution. However, you might notice a drop in the system's potential energy when redistribution occurs.

Initially, the 20.0-\(\mu\text{F}\) capacitor had energy \(E_i = \frac{1}{2} C V^2\). When connected to another capacitor, the total stored energy appears to decrease. This happens because some energy is transferred to heat and electromagnetic radiation during the rapid redistribution of charge.
The calculated initial energy was 6.4 J. After redistribution, the final energy becomes 4.2667 J. The apparent loss of 2.1333 J exemplifies how some energy escapes as heat due to non-ideal conditions in real capacitors, demonstrating energy conversion rather than loss.
Capacitance Calculation
Understanding capacitance and its calculation is essential when dealing with capacitors. Capacitance \(C\) is a measure of a capacitor's ability to store charge per unit voltage, given by \(C = \frac{Q}{V}\).

In our scenario, two capacitors are connected in parallel after the initial capacitor is charged. The total capacitance \(C_{eq}\) is the sum of the capacitances of both capacitors. So, \(C_{eq} = 20.0\, \mu\text{F} + 10.0\, \mu\text{F} = 30.0\, \mu\text{F}\).
  • This illustrates how total capacitance in parallel combinations can be found by simple addition.
  • It shows that parallel capacitors share the total charge across them, leading to the same potential difference \(V_f\) across each.
The resultant final voltage \(V_f\) is calculated using conservation of charge and the formula \(V_f = \frac{Q}{C_{eq}}\). Thus, the understanding of parallel capacitance is vital for solving such problems accurately.

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Most popular questions from this chapter

A parallel-plate capacitor has plates with area 0.0225 m\(^2\) separated by 1.00 mm of Teflon. (a) Calculate the charge on the plates when they are charged to a potential difference of 12.0 V. (b) Use Gauss's law (Eq. 24.23) to calculate the electric field inside the Teflon. (c) Use Gauss's law to calculate the electric field if the voltage source is disconnected and the Teflon is removed.

A cylindrical capacitor has an inner conductor of radius 2.2 mm and an outer conductor of radius 3.5 mm. The two conductors are separated by vacuum, and the entire capacitor is 2.8 m long. (a) What is the capacitance per unit length? (b) The potential of the inner conductor is 350 mV higher than that of the outer conductor. Find the charge (magnitude and sign) on both conductors.

Three capacitors having capacitances of 8.4, 8.4, and 4.2 \(\mu\)F are connected in series across a 36-V potential difference. (a) What is the charge on the 4.2-\(\mu\)F capacitor? (b) What is the total energy stored in all three capacitors? (c) The capacitors are disconnected from the potential difference without allowing them to discharge. They are then reconnected in parallel with each other, with the positively charged plates connected together. What is the voltage across each capacitor in the parallel combination? (d) What is the total energy now stored in the capacitors?

In one type of computer keyboard, each key holds a small metal plate that serves as one plate of a parallel-plate, airfilled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard, the area of each metal plate is 42.0 mm\(^2\), and the separation between the plates is 0.700 mm before the key is depressed. (a) Calculate the capacitance before the key is depressed. (b) If the circuitry can detect a change in capacitance of 0.250 pF, how far must the key be depressed before the circuitry detects its depression?

A 0.350-m-long cylindrical capacitor consists of a solid conducting core with a radius of 1.20 mm and an outer hollow conducting tube with an inner radius of 2.00 mm. The two conductors are separated by air and charged to a potential difference of 6.00 V. Calculate (a) the charge per length for the capacitor; (b) the total charge on the capacitor; (c) the capacitance; (d) the energy stored in the capacitor when fully charged.

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