Figure E24.1424.14 shows a system of four capacitors, where the potential difference across ab is 50.050.0 V. How much charge is stored by this combination of capacitors?

Ch 24: Capacitance and Dielectrics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 24: Capacitance and Dielectrics

Problem 14b

Chapter 24, Problem 14b

Figure E $24.14$ shows a system of four capacitors, where the potential difference across ab is $50.0$ V. How much charge is stored by this combination of capacitors?

Verified step by step guidance

Identify the configuration of the capacitors. Capacitors C2 and C3 are in parallel, and their combination is in series with C1 and C4.

Calculate the equivalent capacitance of the parallel capacitors C2 and C3. Use the formula for capacitors in parallel: C_parallel = C2 + C3.

Calculate the equivalent capacitance of the series combination of C1, C_parallel (from step 2), and C4. Use the formula for capacitors in series: 1/C_series = 1/C1 + 1/C_parallel + 1/C4.

Determine the total charge stored by the combination using the formula Q = C_total * V, where C_total is the equivalent capacitance from step 3 and V is the potential difference across ab (50.0 V).

Ensure all units are consistent and verify each calculation step to ensure accuracy in determining the charge stored by the capacitors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage, measured in farads (F). It is defined by the formula C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the capacitor. In this problem, the individual capacitances of the capacitors (10.0 µF, 5.0 µF, 8.0 µF, and 9.0 µF) are crucial for calculating the total capacitance of the system.

Series and Parallel Capacitors

Capacitors can be connected in series or parallel, affecting the total capacitance of the circuit. In series, the total capacitance (C_total) is given by 1/C_total = 1/C1 + 1/C2 + ... for each capacitor. In parallel, the total capacitance is the sum of individual capacitances: C_total = C1 + C2 + .... Understanding how to combine these capacitors in the given configuration is essential for solving the problem.

Charge Storage in Capacitors

The charge stored in a capacitor is directly proportional to the capacitance and the voltage across it, as described by the equation Q = C × V. In this scenario, with a total capacitance calculated from the configuration and a potential difference of 50.0 V, this relationship will allow us to determine the total charge stored in the combination of capacitors.

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Related Practice

Textbook Question

For the system of capacitors shown in Fig. E $24.16$ , find the equivalent capacitance between $b$ and $c$ .

A parallel-plate air capacitor is to store charge of magnitude $240.0$ pC on each plate when the potential difference between the plates is $42.0$ V.

(a) If the area of each plate is $6.80$ cm², what is the separation between the plates?

(b) If the separation between the two plates is double the value calculated in part (a), what potential difference is required for the capacitor to store charge of magnitude $240.0$ pC on each plate?

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Textbook Question

Figure E $24.14$ shows a system of four capacitors, where the potential difference across ab is $50.0$ V. How much charge is stored in each of the $10.0$ - $μ$ F and the $9.0$ - $μ$ F capacitors?

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Textbook Question

In Fig. E $24.20$ , $C sub 1 equals 6.00$ $μ$ F, $C sub 2 equals 3.00$ $μ$ F, and $C sub 3 equals 5.00$ $μ$ F. The capacitor network is connected to an applied potential $V_{a b}$ .

(a) After the charges on the capacitors have reached their final values, the charge on $C sub 2$ is $30.0$ mC. What are the charges on capacitors $C sub 1$ and $C sub 3$ ?

(b) What is the applied voltage $V_{a b}$ ?

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Textbook Question

A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is $10.0$ pC. The inner cylinder has radius $0.50$ mm, the outer one has radius $5.00$ mm, and the length of each cylinder is $18.0$ cm.

(a) What is the capacitance?

(b) What applied potential difference is necessary to produce these charges on the cylinders?

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Textbook Question

A spherical capacitor contains a charge of $3.30$ nC when connected to a potential difference of $220$ V. If its plates are separated by vacuum and the inner radius of the outer shell is $4.00$ cm, calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

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