Chapter 18, Problem 54P

To determine

The energy required transmitting one potential along the axon with capacitance and the minimum average power required for neurons.

Answer to Problem 54P

The energy required transmitting one potential along the axon with capacitance ${10}^{-8}\text{F}$ is $5\times {10}^{-11}\text{J}$ and the minimum average power required for neurons is $5\times {10}^{-5}\text{J}$ .

Explanation of Solution

Given:

Capacitance of axon, $C={10}^{-8}\text{F}$

Number of neurons, $n={10}^4$

Transmission rate per second, $t=\frac{1}{100}=0.01\text{s}$

Voltage, $V=0.1\text{V}$

Formula used:

Write the expression to calculate the energy required to transmit one potential with axon.

  $E=\frac{1}{2}C{V}^2$.... (1)

Where,

  $V$ is voltage.

Write the expression to calculate the minimum average power required for neurons.

  $P=\frac{E}{t}$ .... (2)

Where,

  $P$ is the minimum average power.

Calculation:

Substitute the values in equation (1).

  $\begin{array}{c}E=\frac{1}{2}\left({10}^{-8}\right){\left(0.1\right)}^2\\ =5\times {10}^{-11}\text{J}\end{array}$

Substitute the values in equation (2).

  $\begin{array}{c}P=\frac{5\times {10}^{-11}\times {10}^4}{0.01}\\ =5\times {10}^{-5}\text{J}\end{array}$

Conclusion:

Therefore, the energy required transmitting one potential along the axon with capacitance ${10}^{-8}\text{F}$ is $5\times {10}^{-11}\text{J}$ and the minimum average power required for neurons is $5\times {10}^{-5}\text{J}$ .