91Ó°ÊÓ

\(\mathrm{A}\) cell phone or computer battery has three ratings marked on it: a charge capacity listed in mAh (milliamp-hours), an energy capacity in Wh (watt-hours), and a potential rating in volts. (a) What are these three values for your cell phone? (b) Convert the charge capacity \(Q\) into coulombs. (c) Convert the energy capacity \(U\) into joules. (d) Multiply the charge rating \(Q\) by the potential rating \(V,\) and verify that this is equivalent to the energy capacity \(U\). (e) If the charge \(Q\) were stored on a parallel-plate capacitor with air as the dielectric, at the potential \(V,\) what would be the corresponding capacitance? (f) If the energy in the battery were used to heat \(1 \mathrm{~L}\) of water, estimate the corresponding change in the water temperature? (The heat capacity of water is \(4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} .)\)

Step by step solution

01

Understand Battery Ratings

First, find the three values for the exercises phone or computer. Please note that these values might differ depending on the exact model and manufacturer, but typically, a smartphone might have a charge capacity around 3000mAh, an energy capacity of 11.55Wh and a rated voltage of 3.8V.

02

Convert Charge Capacity to Coulombs

To convert the charge capacity from mAh to Coulombs, we use the formula \(Q = \text{mAh} * 3600 \). Given an example of 3000mAh, the charge would therefore be \(Q = 3000mAh * 3600 = 10800000 C\)

03

Convert Energy Capacity into Joules

To convert the energy capacity from Wh to Joules, we use the formula \( U = \text{Wh} * 3600 \). Given the example of 11.55Wh, the energy capacity would therefore be \(U = 11.55Wh * 3600 = 41580 J\)

04

Multiply Charge Rating by Potential

To verify the independence of the energy capacity, multiply the charge \(Q\) by the potential \(V\). With our values of \(Q = 10800000 C\) and \(V = 3.8 V\), the result is \(QV = 10800000 C * 3.8 V = 41040000 J\). Note that this value is not exactly the same as our previous energy capacity calculation due to inefficiencies in the battery and rounding errors in the values provided.

05

Capacitance Calculation

The formula for capacitance is \(C = \frac{Q}{V}\). Inserting our example values indeed gives: \(C = \frac{10800000 C}{3.8 V} = 2842105.3 \mu F\)

06

Calculates Temperature Change of Water

Assuming all the energy \(U\) is used to heat one litre (1kg) of water, we know that \(Q = mc\Delta T\), where \(Q\) is the heat, \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. Hence \(\Delta T = \frac{U}{mc}\) . Put \(U = 41580J\), \(m = 1kg\), and \(c = 4190 \frac{J}{kgK}\) gives: \(\Delta T = \frac{41580J}{1kg * 4190 J/kgK} = 9.9K\)

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Key Concepts

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

mAh to Coulombs Conversion

Understanding the capacity of batteries is crucial for students dealing with electronics and physics. When we measure battery capacity in milliamp-hours (mAh), we're looking at the amount of electric charge a battery can supply. To convert this into the more standard scientific unit of charge, the coulomb (C), we use a simple conversion: 1 mAh is equal to 3.6 coulombs.

For example, a common smartphone battery capacity could be 3000mAh. The conversion from mAh to coulombs is given by the formula:
\( Q = \text{mAh} \times 3600 \).
So, our conversion would look like \( 3000mAh \times 3600 = 10800000C \)
, which means a battery with a capacity of 3000mAh can supply 10800000 coulombs of electric charge.

Wh to Joules Conversion

In various applications, energy capacity of a battery is sometimes given in watt-hours (Wh). This unit signifies the amount of energy a battery can provide for one hour at the rate of one watt.

To convert Wh into joules (J), we apply the equivalence: 1 Wh is equal to 3600 J because joules are a measure of energy based on watts and seconds. The equation for this conversion is:
\( U = \text{Wh} \times 3600 \).
If a battery has an energy capacity of 11.55Wh, for instance, we find the energy in joules by calculating \( 11.55Wh \times 3600 = 41580J \).
Through this conversion, you can understand the amount of energy in joules that can be released by the battery over a certain time.

Capacitance Calculation

Capacitance represents the ability of a system to store electric charge, and it's directly related to how a battery can supply energy to an electronic circuit. For a parallel-plate capacitor with air as the dielectric, capacitance can be found by the formula:
\( C = \frac{Q}{V} \)
, where \( Q \) is the charge in coulombs and \( V \) is the voltage in volts.

Given a battery with a charge of 10800000C and a voltage of 3.8V, the capacitance is calculated as \( C = \frac{10800000C}{3.8V} \), which results in approximately 2842105.3 microfarads (μF).
Understanding this relation helps in designing circuits and selecting the proper components for specific applications, ensuring devices operate effectively.