Chapter 26 – The Electric Field

This chapter summarizes how to find the electric field due to a bunch of charges distributed through space.

1. The key idea is that the electric field at some spot is produced by the combined effect of all the charges that cause a field at that spot. This is a vector sum of all the individual electric fields. Each of the contributing charges can be treated as a “point charge.”

2. The electric field due to some small number of charges (2 to 4) can be found by vector addition. Look at the examples in the book to see how this is done.

3. For continuous distributions, rod, infinite wire, ring, disk, or plane, the trick is to set up a summation over the contributions from “small” pieces of the charged object and then to turn that summation into an integral. Study the examples in the text and the ones done in class. The ability to express quantities as sums and then to turn those sums into integrals is very, very useful.

Electric field a distance d from an infinite line of charge, λ = charge/meter, is 2 k λ/d.

Electric field due to an infinite plane of charge, η = charge/meter², η/2ε₀.

4. A parallel plate capacitor is defined as two planes with equal and opposite charges that are close enough to one another that the electric fields inside and outside can be found by using the field due to an infinite plan. The parallel plate capacitor is useful because as defined above, the electric field between the two plates is constant. This makes the space between the plates ideal for studying the behavior of charges moving in a constant electric field. The constant electric field inside a parallel plate capacitor is η/ε₀.

5. The field due to a dipole, two equal but opposite charges, q, separated by a small distance, is calculated using the method described in 1 above. If the separation between the charges is s, the vector dipole moment p is defined as q s, where the vector s points from the negative to the positive charge. Its magnitude is just the distance separating the charges multiplied by q.

Note: Dipoles are important because many molecules act as little atomic dipoles water being a good example.

6. Be able to show why the force on a dipole in a uniform field is zero but the torque on the dipole is given by p x E. Also be able to explain why a dipole in a non-uniform field will experience a net force.

7. It is very useful and important to learn how to check an answer be seeing if your result agrees with some previous answer to a “more simple” problem. Oftentimes the check is done by letting some parameter become a lot bigger or smaller than some other parameter. In those cases, the binomial approximation summarized below is often very valuable when making the comparison.

8. The binomial approximation is used to find the electric field at points far from the charge distribution. The approximation is very important and ought to be filed away for future use, (1 + x)ⁿ ≅1 + n x. This is just a special case of the Taylor Series from your calculus class,

f(x) ≅ f(0) + f’(0) x + ½ f”(0) x² + ...

(Show that the binomial expansion follows from the Taylor series when f(x) = (1 + x)ⁿ

And f’(x) = n(1 + x)ⁿ and f’(0) = n.)