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Chapter 10: Problem 71

A student makes a homemade resistor from a graphite pencil \(5.00 \mathrm{cm}\) long, where the graphite is \(0.05 \mathrm{mm}\) in diameter. The resistivity of the graphite is \(\rho=1.38 \times 10^{-5} \Omega / \mathrm{m} .\) The homemade resistor is place in series with a switch, a 10.00 -mF capacitor and a 0.50 -V power source. (a) What is the \(R C\) time constant of the circuit? (b) What is the potential drop across the pencil 1.00 s after the switch is closed?

Short Answer

Expert verified
After calculating the resistance of the graphite pencil (R), we find the time constant, RC (part a), by multiplying the resistance by the capacitance (C = 10.00 mF). Finally, we use the formula \(V(t) = V_0 \left(1 - e^{-t/RC}\right)\) to calculate the potential drop across the pencil after 1 second (part b), where \(V_0\) is the initial voltage (0.50V). The potential drop across the pencil 1 s after the switch is closed is given by: \[V(1) = 0.50 \left(1 - e^{-1/RC}\right)\]

Step by step solution

01

Calculate the resistance of the graphite pencil (R)

To calculate the resistance of the graphite pencil, we will use the formula: \[R = \frac{\rho L}{A}\] where \(R\) is the resistance, \(\rho\) is the resistivity, \(L\) is the length of the pencil, and \(A\) is the cross-sectional area of the pencil. We are given the resistivity of graphite \(\rho = 1.38 \times 10^{-5} \Omega/\mathrm{m}\) and the length of the pencil \(L = 5.00 cm = 0.0500m\). First, we need to find the cross-sectional area of the pencil. The diameter of the pencil is given as 0.05 mm. To convert it to meters, we multiply by \(1 \times 10^{-3}\): \(d = 0.05 \times 10^{-3}\) m The cross-sectional area of the pencil can be calculated using the formula for the area of a circle: \[A = \pi \Big(\frac{d}{2}\Big)^2\] Plug in the diameter and find the cross-sectional area: \[A = \pi \Big(\frac{0.05 \times 10^{-3}}{2}\Big)^2\] Now calculate the resistance of the pencil: \[R = \frac{\rho L}{A}\]
02

Calculate the time constant, RC (part a)

To find the time constant of the circuit, we need to multiply the resistance by the capacitance: \[RC = R \times C\] We are given the capacitance of the capacitor as \(C = 10.00 mF = 10 \times 10^{-3}F\). Multiply the resistance and the capacitance to find the time constant: \[RC = R \times (10 \times 10^{-3})\]
03

Calculate the potential drop across the pencil after 1 second (part b)

To find the potential drop across the pencil resistor 1 second after the switch is closed, we will use the formula for the voltage in an RC circuit after time t: \[V(t) = V_0 \left(1 - e^{-t/RC}\right)\] where \(V(t)\) is the voltage at time \(t\), \(V_0\) is the initial voltage, and \(RC\) is the time constant. We are given the initial voltage as \(V_0 = 0.50V\), and we must find the voltage 1 second after the switch is closed: \(t = 1.00 s\). Plug in the values and calculate the potential drop across the pencil resistor: \[V(1) = 0.50 \left(1 - e^{-1/RC}\right)\]

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Key Concepts

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

Resistor
In an RC circuit, a resistor is a crucial component that determines how quickly the capacitor charges and discharges. A resistor opposes the flow of current, causing a drop in the voltage across it.
This resistance is especially important in controlling the timing characteristics of the circuit, as explained by the time constant.
A resistor is defined by its resistance value, measured in ohms (惟), and can be calculated if the material's resistivity, length, and cross-sectional area are known.
For example, to create a homemade resistor out of a graphite pencil, one must consider the pencil's length and diameter.
The resistivity of graphite, along with these measurements, enables us to determine the total resistance using the formula:
  • \[ R = \frac{\rho L}{A} \]
  • \(R\) is the resistance.
  • \(\rho\) is the resistivity of the material.
  • \(L\) is the length.
  • \(A\) is the cross-sectional area.
Ultimately, understanding how to calculate resistance is vital for analyzing how a resistor interacts with a capacitor in an RC circuit.
Capacitor
A capacitor in an RC circuit temporarily stores electrical energy in an electric field between its plates.
It's crucial for controlling the timing and smoothing out electrical signals in circuits.
In such circuits, a capacitor is specified by its capacitance value, measured in farads (F). This indicates how much charge it can store for a given voltage.
Capacitors respond to the voltage applied across them, charging up and then releasing the stored charge when necessary.
The relationship between the capacitor and the resistor in the circuit is essential to the circuit鈥檚 function.
  • The time it takes for a capacitor to charge or discharge to roughly 63% of its total charge is determined by the time constant \((RC)\).
  • This makes capacitors a critical component in timing-based applications.
When the switch in an RC circuit is closed, the capacitor begins to charge through the resistor, with the speed determined by their respective values.
Graphite Resistivity
Graphite is used in homemade resistors due to its known resistivity, making it a practical choice for DIY electronics.
Resistivity is a measure of how strongly a material opposes the flow of electric current.
For graphite, the resistivity is given as
  • \(\rho = 1.38 \times 10^{-5} \Omega/m\).
This low resistivity indicates that graphite is a relatively good conductor of electricity.
It's less resistant than many other materials used in electronic components.
To use graphite as a resistor, the pencil's dimensions and its resistivity value allow us to determine the total resistance it provides.
These measurements directly affect how the RC circuit behaves, especially in applications where precise resistance is necessary.
Understanding this concept is key to effectively using graphite and similar materials in educational and hobbyist circuits.
Time Constant
The concept of a time constant
  • \(\tau = RC\)
is fundamental to RC circuits. It predicts how quickly the voltage across the capacitor will rise or fall once the circuit is active.
Being able to calculate and understand the time constant is critical in designing circuits that require precise timing.
The time constant \((RC)\) is calculated by multiplying the resistance \((R)\) from the resistor and the capacitance \((C)\) of the capacitor, both of which are in the circuit.
This product indicates how quickly the capacitor can charge through the resistor.
You can think of the time constant as the number of seconds it takes for the capacitor to charge up to about 63% of the applied voltage, or to discharge to about 37% when the circuit is powered down.
It's crucial for predicting the behavior of circuits in various applications, from simple timers to more complex devices.

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Most popular questions from this chapter

A \(120-\mathrm{V}\) immersion heater consists of a coil of wire that is placed in a cup to boil the water. The heater can boil one cup of \(20.00^{\circ} \mathrm{C}\) water in 180.00 seconds. You buy one to use in your dorm room, but you are worried that you will overload the circuit and trip the \(15.00-\mathrm{A}\) 120-V circuit breaker, which supplies your dorm room. In your dorm room, you have four \(100.00-\mathrm{W}\) incandescent lamps and a \(1500.00-\mathrm{W}\) space heater. (a) What is the power rating of the immersion heater? (b) Will it trip the breaker when everything is turned on? (c) If it you replace the incandescent bulbs with 18.00-W LED, will the breaker trip when everything is turned on?

A \(160-\mu \mathrm{F}\) capacitor charged to \(450 \mathrm{V}\) is discharged through a \(31.2-\mathrm{k} \Omega\) resistor. (a) Find the time constant. (b) Calculate the temperature increase of the resistor, given that its mass is \(2.50 \mathrm{g}\) and its specific heat is \(1.67 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) noting that most of the thermal energy is retained in the short time of the discharge. (c) Calculate the new resistance, assuming it is pure carbon. (d) Does this change in resistance seem significant?

We are often advised to not flick electric switches with wet hands, dry your hand first. We are also advised to never throw water on an electric fire. Why?

A \(500-\Omega\) resistor, an uncharged \(1.50-\mu \mathrm{F}\) capacitor, and a \(6.16-\mathrm{V}\) emf are connected in series. (a) What is the initial current? (b) What is the \(R C\) time constant? (c) What is the current after one time constant? (d) What is the voltage on the capacitor after one time constant?

The severity of a shock depends on the magnitude of the current through your body. Would you prefer to be in series or in parallel with a resistance, such as the heating element of a toaster, if you were shocked by it? Explain.
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