Problem 36 - Physics

Question

An axon is the relatively long tail-like part of a neuron, or nerve cell. The outer surface of the axon membrane (dielectric constant \(=5,\) thickness \(=1 \times 10^{-8} \mathrm{~m}\) ) is charged positively, and the inner portion is charged negatively. Thus, the membrane is a kind of capacitor. Assuming that an axon can be treated like a parallel plate capacitor with a plate area of \(5 \times 10^{-6} \mathrm{~m}^{2},\) what is its capacitance?

Step-by-Step Solution

Answer

The capacitance of the axon is approximately \( 2.2125 \times 10^{-11} \) Farads.

Step 1: Understand the formula for capacitance of a parallel plate capacitor

The formula to calculate the capacitance \( C \) of a parallel plate capacitor is given by \( C = \frac{\varepsilon_0 \times \varepsilon_r \times A}{d} \), where \( \varepsilon_0 \) is the vacuum permittivity \( (8.85 \times 10^{-12} \, \text{F/m}) \), \( \varepsilon_r \) is the dielectric constant, \( A \) is the area of the plates, and \( d \) is the separation between the plates (thickness of the dielectric).

Step 2: Substitute the given values

Given, the dielectric constant \( \varepsilon_r = 5 \), the thickness of the dielectric \( d = 1 \times 10^{-8} \, \text{m} \), and the area \( A = 5 \times 10^{-6} \, \text{m}^2 \). Substitute these values into the formula: \( C = \frac{8.85 \times 10^{-12} \times 5 \times 5 \times 10^{-6}}{1 \times 10^{-8}} \).

Step 3: Perform the calculation

Perform the multiplication and division: \( C = \frac{8.85 \times 10^{-12} \times 25 \times 10^{-6}}{1 \times 10^{-8}} = \frac{221.25 \times 10^{-18}}{1 \times 10^{-8}} = 22.125 \times 10^{-12} \). Simplify the result to \( C = 2.2125 \times 10^{-11} \) F.

Step 4: Interpret the result

The calculated capacitance of the axon, treated as a parallel plate capacitor, is \( 2.2125 \times 10^{-11} \) Farads.

Key Concepts

Parallel Plate Capacitor

A parallel plate capacitor is an electrical component composed of two conductive plates separated by a layer of insulating material, which is called a dielectric. When a voltage is applied across the plates, opposite charges build up on the plates, creating an electric field between them. This structure allows the capacitor to store electrical energy and release it when needed.
Understanding the basic characteristics of parallel plate capacitors will help you grasp how these devices function in circuits:

• **Plate area (\(A\))**: The capacitance of the capacitor is directly proportional to the area of the plates. A larger plate area provides a greater capacity to store charge.
• **Distance between plates (\(d\))**: The capacitance is inversely proportional to the distance between the two plates. If the plates are closer, the capacitance increases as the electric field is stronger.
• **Material between plates**: The dielectric material inserted between the plates strongly influences the capacitance, described by its dielectric constant.

The formula to calculate capacitance (\(C\)) is: \[C = \frac{\varepsilon_0 \times \varepsilon_r \times A}{d} \]where \(\varepsilon_0\) is the vacuum permittivity, and \(\varepsilon_r\) is the dielectric constant of the material. This expression clearly highlights how capacitance is affected by its geometric and material properties.

Dielectric Constant

The dielectric constant (often represented as \(\varepsilon_r\)) is a crucial factor in determining the capacitance of a parallel plate capacitor. It defines how well a material can store electrical energy in an electric field compared to a vacuum.
Here's what you need to know:

• **High dielectric constant materials**: These improve the capacitance of a capacitor significantly by reducing the strength of the electric field inside the dielectric. Materials with a high dielectric constant can store more energy and are often used when you need a compact capacitor with high capacitance.
• **Role in capacitors**: In the context of capacitors, the dielectric not only separates the conductive plates but also contributes to the capacitor's storage properties by increasing capacitance.
• **Symbol and measurement**: The dielectric constant is a unitless ratio, meaning it has no units, being understood as a relative comparison to the vacuum.

For example, if a material has a dielectric constant of 5, as in the case of the axon membrane, it means this material can "boost" the capacitance fivefold compared to the same setup with air or vacuum as the dielectric.

Neuron

Neurons are fascinating cells that form the fundamental working units of the brain, nervous system, and spinal cord. They are crucial for transmitting signals throughout your body and consist of several parts, each serving a unique function.
A typical neuron includes:

• **Dendrites**: Structures that receive chemical signals from the axons of other neurons.
• **Cell body (soma)**: Contains the nucleus and is vital for maintaining the cell and its functions.
• **Axon**: A long, tail-like strand that conducts electrical impulses away from the neuron's cell body to other neurons, muscles, or glands.

In our specific context, the axon can be likened to a capacitor, having a charged membrane that separates positive and negative charges. This membrane's function can be compared to a parallel plate capacitor, with its dielectric constant impacting how it stores electrical charge.
This analogy to parallel plate capacitors helps explain how neurons might manage to convey signals – by temporarily storing and quickly releasing electrical charges, facilitating the rapid transmission of signals between neurons.