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Chapter 23: Problem 59

A cubical region \(1.0 \mathrm{m}\) on a side is located between \(x=0\) and \(x=1 \mathrm{m} .\) The region contains an electric field whose magnitude varies with \(x\) but is independent of \(y\) and \(z: E=E_{0}\left(x / x_{0}\right),\) where \(E_{0}=24 \mathrm{kV} / \mathrm{m}\) and \(x_{0}=6.0 \mathrm{m} .\) Find the total energy in the region.

Short Answer

Expert verified
The total energy in the cubical region is 0.106 kJ.

Step by step solution

01

Write Down the Energy Density Function

The energy density \(u\) in an electromagnetic field is given by \(u = \frac{1}{2}ε_0E^2\). Here, \(E\) is given by \(E=E_{0}*\frac{x}{x_{0}}\), where \(E_0 = 24 \, kV/m = 24 * 10^3 \, V/m\) and \(x_{0}=6.0 \, m\). Therefore, \(u(x) = \frac{1}{2}\epsilon_0\left(\frac{E_0x}{x_0}\right)^2\).
02

Integrate Over The Volume

The total energy \(U\) is obtained by integrating the energy density over the volume V of the cubical region. We integrate over \(x\) from 0 to 1, over \(y\) from 0 to 1, and over \(z\) from 0 to 1. Therefore, \(U = \int_0^1 \int_0^1 \int_0^1 u(x) dxdydz\). Since \(u(x)\) does not depend on \(y\) and \(z\), we can separate the integrals: \(U = \int_0^1 dx \int_0^1 dy \int_0^1 dz * u(x)\).
03

Evaluate The Integral

Perform the integration over \(x\), \(y\), and \(z\). The integrations over \(y\) and \(z\) each give 1, because these are constants from 0 to 1, while the integral over \(x\) will yield the value for the total energy in the cube. Assemble all constants outside the integral sign and perform the integration: \(U = \epsilon_0 * \left(\frac{E_0}{x_0}\right)^2 * \int_0^1 x^2 dx = \epsilon_0 * \left(\frac{E_0}{x_0}\right)^2 * \frac{1}{3}x^3|_0^1\).
04

Calculate the Total Energy

Substitute the given values: \(E_0 = 24*10^3 \, V/m\) , \(x_0 = 6.0 \, m\), and \(\epsilon_0 = 8.85*10^{-12} C^2/N*m^2\) into the equation for total energy \(U\), which gives \(U = 8.85*10^{-12} C^2/N*m^2 * (24*10^3 V/m / 6.0 m)^2*1/3 = 106 J = 0.106 kJ\).

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Most popular questions from this chapter

How does the energy density at a certain distance from a negative point charge compare with the energy density at the same distance from a positive point charge of equal magnitude?

A medical defibrillator stores \(950 \mathrm{J}\) in a \(100-\mu \mathrm{F}\) capacitor. (a) What is the voltage across the capacitor? (b) If the capacitor discharges \(300 \mathrm{J}\) of its stored energy in \(2.5 \mathrm{ms}\), what's the power delivered during this time?

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