Departamento de física – Carrera de física

Problemas de electricidad y magnetismo Lista 5

(Tiempo para resolver esta lista semana y media)

Segundo semestre de 2013

1. Calcule el momento dipolar, el potencial eléctrico y el campo eléctrico de la siguiente

distribución de carga

Si sobre esta distribución de carga de aplica un campo eléctrico E = 2105N/Cj,

calcule:

- La fuerza dipolar

- La energía potencial dipolar

- El torque sobre el dipolo

Para d = 0.1cm, = 30º y q = 2C

2. Un capacitor es formado por dos cilindros conductores concéntricos de radios a y b

(a<b) y longitud L (L>>b). Considerando que entre ellos hay una diferencia de potencial

V resuelva la ecuación de Laplace. Encuentre el campo eléctrico, la energía almacenada

en el condensador como dVEU 20

2

. A partir de este resultado halle la capacitancia

del sistema.

3. Let 𝜑(𝑥, 𝑦, 𝑧) be any function that can be expanded in a power series around a point

(𝑥𝑜 , 𝑦𝑜 , 𝑧𝑜). Write a Taylor series expansion for the values of 𝜑 at each the six points

(𝑥𝑜 + 𝛿, 𝑦𝑜 , 𝑧𝑜). (𝑥𝑜 − 𝛿, 𝑦𝑜 , 𝑧𝑜), (𝑥𝑜 , 𝑦𝑜 + 𝛿, 𝑧𝑜), (𝑥𝑜 , 𝑦𝑜 − 𝛿, 𝑧𝑜), (𝑥𝑜 , 𝑦𝑜 , 𝑧𝑜 + 𝛿) and

(𝑥𝑜 , 𝑦𝑜 , 𝑧𝑜 − 𝛿), which symmetricall surround the point (𝑥𝑜 , 𝑦𝑜 , 𝑧𝑜) at a distance 𝛿. Show

that if 𝜑 satisfies the Laplace’s equation, the average of the six values is equal to

𝜑(𝑥𝑜 , 𝑦𝑜 , 𝑧𝑜) through terms of third order in 𝛿.

4 (3.30). Here’s how to solve Laplace’s equation approximately, for given boundary

values, using nothing but arithmetic. The method is the relaxation method (see section 3.8

of Pourcell), and it is based on the result of problem 3. For simplicity we take a two-

dimensional example. In the figure there are two square equipotential boundaries, one

inside the other. This might be a cross section thorough a capacitor made of two sizes of

metal tubing. The problem is to find, for an array of discrete points, numbers which will be

z

y

q

-q

d

3)

good approximation to the values at those points of the exact two-dimensional potential

function 𝜑(𝑥, 𝑦). For this exercise, we’ll make the array rather coarse, to keep the labor

within bonds. Let us assign, arbitrarily, potential to 100V to the inner boundary and zero

the outer. All points on these boundaries retain those values. You could start with any

values at the interior points, but time will be saved by a little judicious guesswork. We

know the correct values must be lie between 0V and 100V, and we expect that points closer

to the inner boundary will have higher values than those closer to the outer boundary. Some

reasonable starting values are suggested in the figure. Obviously, you should take

advantage of the symmetry of the configuration: Only seven different interior values need

to be computed. Now you simply go over these seven interior lattice points in some

systematic manner.

5. Three conducting plates are placed parallel to one another as shown. The outer plates are connected by a wire. The inner plate is isolated and carries a charge

amounting to 10esu per square centimeter of plate. In what proportion must this charge divide itself into a surface charge 𝛔𝟏on one face of the inner plate and a surface charge 𝜎2 on the other side of the same plate?

6. What is the capacitance C of a capacitor that consist of two concentric spherical shaells? The inner radius of the outer shell is a; the outer radius of the inner shell is b. Check your result by considering the limiting case with the gap between the conductors, a-b, much smaller than b. In that limit the formula for the capacitance of the flat parallel-plate capacitor ought to be applicable. 7. 100 pF capacitor is charged to 100 V. After charging battery is disconnected, the capacitor is connected in parallel to another capacitor. If the final voltage is 30V, what is the capacitance of the second capacitor? How much energy was lost, and what happened to it? 8.