Problem 36 - College Physics

Question

(a) You find that if you place charges of \(\pm 1.25 \mu \mathrm{C}\) on two separated metal objects, the potential difference between them is 11.3 \(\mathrm{V}\) . What is their capacitance? (b) A capacitor has a capacitance of 7.28\(\mu \mathrm{F}\) . What amount of excess charge must be placed on each of its plates to make the potential difference between the plates equal to 25.0 \(\mathrm{V}\) ?

Step-by-Step Solution

Answer

(a) 110.6 nF; (b) 182 μC.

Step 1: Understanding Capacitance

To find the capacitance, we use the formula \( C = \frac{Q}{V} \), where \( Q \) is the charge and \( V \) is the potential difference. This formula defines capacitance as the charge stored per unit voltage.

Step 2: Calculate Capacitance for Given Charges and Voltage

Given \( Q = 1.25 \mu C = 1.25 \times 10^{-6} C \) and \( V = 11.3 V \), substitute into the formula: \[C = \frac{1.25 \times 10^{-6}}{11.3}\]Calculate to find:\[C \approx 1.106 \times 10^{-7} F = 110.6 nF\]

Step 3: Understanding Charge in Capacitors

For part (b), we need to find the charge \( Q \) needed for a given capacitance and potential difference. Use \( Q = C \times V \).

Step 4: Calculate Charge for Given Capacitance and Voltage

Given \( C = 7.28 \mu F = 7.28 \times 10^{-6} F \) and \( V = 25 V \), substitute into the formula: \[Q = 7.28 \times 10^{-6} \times 25\]Calculate to find:\[Q = 1.82 \times 10^{-4} C = 182 \mu C\]

Key Concepts

Electric Charge

Electric charge is a fundamental property of matter. It can be either positive or negative. Objects can attract or repel each other based on their charges.
When you have a positive charge and a negative charge, they attract each other, like magnets do. Similarly, like charges (positive with positive, negative with negative) repel each other.

• Measured in coulombs (C).
• Smaller charges are often given in microcoulombs (\(\mu C\)), where 1 \(\mu C = 10^{-6} C\).

Charge plays a critical role in the creation of electric fields. These fields are responsible for the forces exerted by and on charged objects.
Studying how charges interact is essential to understanding capacitors, which store and release these charges.

Potential Difference

Potential difference, also known as voltage, is the work required to move a charge from one point to another in an electric field.
It's measured in volts (V) and represents the energy difference per charge between two points.

• A high potential difference means lots of energy is available for moving charges.
• A low potential difference means less available energy.

Think of potential difference like the pressure in a water hose. Higher pressure is similar to higher voltage, pushing charges more forcefully.
In a capacitor, the potential difference exists between its two plates, resulting in the storage of electric energy.

Capacitor

A capacitor is a device that stores electric charge and energy in an electric field. It consists of two conductors separated by an insulator.
These conductors are often referred to as plates. When connected to a power source, one plate collects positive charge while the other collects an equal negative charge.

• The insulator prevents charges from moving between plates.
• Capacitors are widely used in electronic circuits for energy storage.

Capacitors release stored energy quickly, making them ideal for things like camera flashes and tuning radios.
The most important property of a capacitor is its capacitance, determined by the size and distance of its plates and the nature of the insulating material.

Capacitance Calculation

Capacitance (\(C\)) is a measure of a capacitor's ability to store charge per unit voltage. It is expressed in farads (F), which can also be microfarads (\(\mu F\) or \(10^{-6} F\)).
The formula to calculate capacitance is:
\[C = \frac{Q}{V}\]
where \(Q\) is the charge in coulombs and \(V\) is the potential difference in volts.

• To find capacitance, know the charge and voltage values.
• Rearranging the formula helps find unknown values such as charge or potential difference.

In practical calculations, it's crucial to convert units when necessary (e.g., \(\mu C\) to C or \(\mu F\) to F).
This ensures correct results, such as finding the capacitance in exercise solutions or determining how much charge a capacitor can store.