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Chapter 23: Problem 5

Does the superposition principle hold for electric-field energy densities? That is, if you double the field strength at some point, do you double the energy density as well?

Short Answer

Expert verified
No, the superposition principle does not hold for electric-field energy densities. Doubling the field strength results in a four times increase in the energy density, not double.

Step by step solution

01

Understand Key Concepts

The energy density of an electric field (\(u\)) is given by \(u = 0.5*ε_0*E^2\), where \(E\) is the electric field strength and \(ε_0\) is the permittivity of free space. The superposition principle states that the net response at a given place and time caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.
02

Apply the Superposition Principle

The superposition principle indicates that if we double the field strength (\(2E\)), we should double the energy density as well. Let's substitute \(2E\) into the energy density equation and simplify: \(u = 0.5*ε_0*(2E)^2 = 2*0.5*ε_0*E^2\). This proves that doubling \(E\), actually increases \(u\) fourfold, not twofold.
03

Conclusion

From the above calculation, it is evident that the superposition principle doesn't hold for the electric field energy densities. Doubling the field strength results in a quadrupling of the energy density, not just a doubling.

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Most popular questions from this chapter

A dipole consists of two equal but opposite charges. Is the total energy stored in the dipole's electric field zero? Why or why not?

Your company is still stuck with those 2 - \(\mu\) F capacitors from Problem 50. They turn out to be so cheap that their capacitances are all too low, ranging from \(1.7 \mu \mathrm{F}\) to \(1.9 \mu \mathrm{F}\). A colleague suggests you put variable "trimmer" capacitors in parallel with the cheap capacitors and adjust the combination to precisely \(2.00 \mu \mathrm{F} .\) The available trimmers have variable capacitance from \(25 \mathrm{nF}\) to \(350 \mathrm{nF}\). Will they work?

A classical view of the electron pictures it as a purely electric entity, whose Einstein rest mass energy, \(E=m c^{2},\) is the energy stored in its electric field. If the electron were a sphere with charge distributed uniformly over its surface, what radius would it have in order to satisfy this condition? (Note: Your answer, and the picture of the electron as a sphere, aren't consistent with quantum theory.)

A capacitor's plates hold \(1.3 \mu \mathrm{C}\) when charged to \(60 \mathrm{V}\). What's its capacitance?

Two positive point charges are infinitely far apart. Is it possible, using a finite amount of work, to move them until they're a small distance \(d\) apart?
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