Problem 37 - Single Variable Calculus

Question

Child-Langmuir Law In a vacuum diode a steady current \(I\) flows between the cathode with potential 0 and anode which is held at a positive potential \(V_{0}\). The Child-Langmuir Law states that \(I=k V_{0}^{3 / 2}\), where \(k\) is a constant. Use differentials to estimate the percentage change in the current corresponding to a \(10 \%\) increase in the positive potential.

Step-by-Step Solution

Answer

A 10% increase in the positive potential will result in a 15% change in the current, based on the Child-Langmuir Law and using differentials to estimate the percentage change.

Step 1: Differentiate the given equation with respect to \(V_0\)

We first differentiate the given equation with respect to \(V_0\): \(\frac{dI}{dV_0} = \frac{3}{2}kV_0^{\frac{1}{2}}\) #Step 2: Calculate the change in positive potential#

Step 2: Find the change in positive potential, \(\Delta V_0\)

To find the change in positive potential, we need to multiply the given percentage change by the initial positive potential. Since we do not know the initial potential, we will define \(\Delta V_0\) as a function of the initial potential: \(\Delta V_0 = 0.1V_0\) where the factor 0.1 is the 10% increase in potential. #Step 3: Calculate the change in current using differentials#

Step 3: Calculate the change in current, \(\Delta I\), using differentials

Now we can use the result from Step 1 to find the change in current based on the change in positive potential. We multiply the derivative of \(I\) by \(\Delta V_0\): \(\Delta I = \frac{dI}{dV_0} \cdot \Delta V_0 = \frac{3}{2}kV_0^{\frac{1}{2}} \cdot 0.1V_0\) Simplify and factor the current \(I\) out of the equation, as follows: \(\Delta I = 0.1 \cdot \frac{3}{2}V_0I = 0.15V_0I\) #Step 4: Calculate the percentage change in current#

Step 4: Find the percentage change in current

Finally, we can find the percentage change in current by dividing the calculated change in current, \(\Delta I\), by the initial current, \(I\), and multiplying by 100: Percentage change in current = \(\frac{\Delta I}{I} \cdot 100 = \frac{0.15V_0I}{I} \cdot 100 = 15\%\) Therefore, a 10% increase in the positive potential will result in a 15% change in the current.

Key Concepts

Vacuum Diode

A vacuum diode is an electronic device consisting of two electrodes, a cathode and an anode, separated by a vacuum. The cathode is typically heated, emitting electrons that are attracted to the positively charged anode, creating an electric current. This simple two-electrode system allows for the conduction of electricity in one direction.

Understanding a vacuum diode is essential as it illustrates the basic principles of electron flow and potential difference. It is crucial in tube-based electronics, serving as the foundation for more complex devices like vacuum tubes and semiconductor diodes.

• In this device, no current flows when the anode is negatively charged with respect to the cathode.
• The basic operation of vacuum diodes helps explain important electronic laws such as the Child-Langmuir Law.
• It provides insight into electron tube operations, which were pivotal in the development of early electronics.

Differentiation

Differentiation is a fundamental concept in calculus used to find the rate at which one quantity changes compared to another. In the context of the Child-Langmuir Law, differentiation helps us understand how the electric current changes when the potential difference in a vacuum diode is varied.

During differentiation, we are finding the derivative of a function. For the Child-Langmuir Law equation \( I = kV_0^{3/2} \), differentiating with respect to \( V_0 \) results in: \[ \frac{dI}{dV_0} = \frac{3}{2}kV_0^{\frac{1}{2}} \]

• The derivative \( \frac{dI}{dV_0} \) tells us how much the current \( I \) increases for a small change in voltage \( V_0 \).
• It is a crucial step for calculating changes in variables when dealing with physical laws and equations.

Differentiation thus allows us to practically apply theoretical laws, aiding in predicting the behavior of circuits under varying conditions.

Percentage Change

Percentage change is a way of expressing a change in a quantity as a fraction of its initial value, presented as a percentage. To calculate percentage changes in scientific problems, such as in the Child-Langmuir Law, we must understand the relation between potential change and resulting current change in a vacuum diode.

In practical terms, let's say the potential difference \( V_0 \) is increased by 10%. This translates mathematically into:

• \( \Delta V_0 = 0.1V_0 \), indicating a 10% increase in the positive potential.
• Affects the current \( I \) according to the derivative: \( \Delta I = \frac{3}{2}kV_0^{\frac{1}{2}} \cdot 0.1V_0 \).

Understanding percentage changes aids in analyzing how substantial or negligible a change in input can affect the output result, an important skill in both academic exercises and real-world applications.

Potential Difference

Potential difference, often termed voltage, is the measure of electric potential energy between two points in a circuit. In a vacuum diode, it plays a crucial role in determining the flow of electrons from the cathode to the anode.
The potential difference \( V_0 \) in a diode is vital for creating the electric field that propels electrons, resulting in electrical current. In the Child-Langmuir Law, it is this potential difference that influences the current flowing through the diode, following the equation \( I = kV_0^{3/2} \).

• A higher potential difference typically causes a higher current flow, provided other conditions remain constant.
• The relationship can be analyzed through differentials to assess sensitivity to changes in voltage.
• Understanding potential difference is key to controlling and predicting the behavior of electrical devices such as diodes.

In simple terms, the potential difference is the driving force that enables the vacuum diode to function as intended, facilitating the flow of current through the generated electric field.