chapter23

= **﻿Chapter 23: Electric Potential** = = ﻿Section 1 - Potential Difference = =Electric potential energy is very easy to understand. The concept is more or less the same as gravitational potential energy, with a few small differences. In case you don't remember, gravitational potential energy depends upon an object's mass, strength of gravity, and distance from the source of gravity, usually Earth. An object's electric potential energy depends on the object's charge, the strength of the electric field, and the object's distance from the source of the electric field. As an object moves away from the gravity source, or against the force of gravity, it gains gravitational potential energy. Similarly, as a charged object moves against the force of an electric field, it gains potential energy. However, it is important to understand that moving an object away from an electric field source does not necessarily increase its electric potential energy. This is only true if the the object and source of the electric field are opposite signs. If both are the same sign, moving the object closer to the source of the electric field will increase it's electric potential energy. If you can remember these two scenarios easily, great. Otherwise, just remember that __moving against the force of an electric field will increase an object's electric potential energy.__ = =The equation for calculating electric potential energy is:=

=where k is equal to **8.988 * 10 ^ 9, q is equal to** the charge of one object in Coulombs*, q0 is equal to the charge of the other object in Coulomb's, and r is equal to the distance between the two objects in meters.= =*For example, 14nC would be written as 14 * 10 ^ -9 C=

= ﻿Section 2 - Potential Due to a System of Point Charge = = ﻿To calculate the electric potential energy, use the following equation: = == = Again we have a situation similar to gravitational potential energy, in which an enormous distance between two particles will result in the electric potential energy becoming nearly zero. Just as with gravitational potential energy, even though the electric potential energy is technically never quite zero (there will always be some tiny bit of attraction/repulsion) at a certain point the potential energy becomes negligible. In other words, at some point the electric potential energy between two particles will become so small that its effect will go unnoticed, so we can treat the system as if the potential energy is zero. = = = = But what if you have more than two point charges in your system? Well, you essentially use the same equation as you would for a single point charge, with a small difference. You simply calculate the electric potential caused by each individual point charge and add them together: = == = Section 3 - Computing the Electric Field From the Potential = =Remember that electric potential increases as a charge moves against the force of an electric field. We can actually use electric potential to calculate electric field. Consider the path of this particle to be a vector. If this vector points directly against the force of a scalar function, or exactly in the opposite direction of a scalar function and has magnitude equal to the derivative of this scalar funtion, it is called a __**gradient**__ **.** **Therefore, electric field is equal to the negative gradient of the electric potential. Now, this sounds very confusing, and if you didn't follow my wording take a look a page 724 in the book for a slightly different explanation. However, it is important that you understand the following concepts and equations:**= =**If the electric potential is dependent only on the movement of a particle along one axis, the other two axes will be unaffected by any movement of the particle. Therefore you can set the electric field vectors along these axes equal to zero. So if your particle is moving in the x direction, you would use this equation:**= == =If your particle is moving along another one the axes, simply swap the x's with whatver you need, and the i^ with j^ for y, or with k^ for z. If your electric field is spherical, replace the x in the above equation with r, as in the radius of the spherical field. To put it in the simplest of terms, just remember that electric field is equal to the negative derivative of electric potential. If you have the equation for one, it should be fairly straightforward to get the equation for the other.= = Section 4 - Calculations of V for Continuous Charge Distributions = = Let's say we want to calculate the electric potential of a continuous charge distribution (something like a ring, plane, line, or sphere). Continuous charge distribution means that the charge is spread evenly throughout the object. To calculate the electric potential, we first need to choose a piece of this object which we will designate as //dq// and treat as a point charge. What were are essentially doing is calculating the electric potential of this point charge, then adding up multiple //dq'//s until we have the electric potential. However, instead of using the equation for adding point charges from section 2, we will use this equation: = == = IMPORTANT!!! You cannot use this equation for an infinite line or plane, as it assumes that electric potential is zero at an infinite distance away from the point charge. = =Now I'll just go through the different continuous charge distributions you need to know, in the order they appear in the book. It is probably easiest to just remember these formulas so you can easily calculate electric potential when you encounter a continuous charge distribution.= == =For a ring with a continuous charge with radius a and charge Q, to calculate the electric potential at a point P from= =the center of the ring, use:=

=where r is the hypotenuse of a right triangle formed by a, x, and r.= =For a uniformly charged disk with radius a and charge Q, using the same P and r, use this formula:= == =If you really feel up to the challenge take a look at page 728 in the book to see how this formula is derived, but you will never need to do that on a test.= = = =On to infinite lines and planes. Now, it is important to realize that there is no such thing as a truly infinite plane or line, but we treat very large planes and lines as if they were infinite. For example, if you could treat a massive wall as an infinite plane, or a very long wire as an infinite line (by very long or massive I mean at least a few hundred meters).= =As I stated above in red, we encounter a small problem in dealing with infinite planes or lines because our equation has the potential equal to zero at infinity. We need to make the electric potential zero at some definable point "before" infinity. So first we find the electric field using any method you'd like (see Chapter 22). Then, remembering that electric field is defined as= == =we can can calculate the potential function. Alternatively, you can just remember this formula for infinite planes of charge:= =Again, you do not need to know how to derive this formula for tests, so don't worry if you don't get the derivation in the book. As long as you know how to use the formula, you're all set.= =For a spherical shell of charge, the electric potential calculation changes depending on whether or not your point of interest is inside or outside the sphere. If it is outside or on the sphere, you use this formula:= == =The derivation of this formula can be found on page 731, but it is just integrating the electric field equation, similar to the previous formula's derivation. If your point of interest is inside the sphere, no calculation is needed. Electric potential is zero anywhere inside a spherical shell of charge. As the book wisely points out, don't think that this is true because the electric field is zero inside the spherical shell of charge. Zero electric field only means that the electric potential is the same throughout that region, not that it is necessarily zero. = == =For an infinite line of charge, use this formula:= =where R ref is a reference point greater than zero but less than infinity. The electric potential is zero at this reference point.= = Section 5 - Equipotential Surfaces = =What is an equipotential surface? Pretty much exactly what it sounds like. In a conductor in which the electric field is zero, the electric potential is the same throughout. It's surface is known as an equipotential surface, and the interior is known as an equipotential region. Because the electric potential will be zero for any displacement parallel to the surface, the electric field must be normal to the equipotential surface. In other words, any electric field lines starting or ending at an equipotential surface must be perpendicular to it.= =This chapter also discusses the phenomenon known as dielectric breakdown. Dielectric breakdown is simply when a very strong electric field causes a nonconducting object to become a conductor, and the magnitude of this electric field is known as dielectric strength.= =This section of Chapter 23 seems pretty long, but you won't be tested on most of it. Know the above information and read the section, but you don't need to memorize how a Van de Graaff Generator works.=

= ** Here is a lab you could do in class: ** = =** Objective: **= == Calculate the electric potential in the two dimensional region around 1, 2, 3, or 4 small, uniformly charged spheres. The region is divided into a 25x25 grid. The upper left corner of the grid corresponds to x equals 0.5m, y equals 0.5m, and the lower right corner corresponds to x equals 24.5m, y equals 24.5m. The charged spheres can be placed anywhere on the gird. They will be located in the x-y plane. Students will calculate the potential at each grid point and construct a surface and a contour plot of the potential.== = = =** Procedure: **= = 1. Open a new Microsoft Excel spreadsheet. = = = = 2. Enter the numbers 0.5 to 24.5 in increments of 1 in cells B1 - Z1. Next, enter the numbers 0.5 to 18.5 and 20 to 25 in increments of 1 into the cells A2 - A2. = = = = 3. Copy the following information into your excel chart exactly as it is shown here. = = =

= This is where the magnitude and location of the charges will be determined. = = = = 4. Enter this equation into B2: =9*$C$31/SQRT((B$1-$A$31)^2+($A2-$B$31)^2) +9*$C$32/SQRT((B$1-$A$32)^2+($A2-$B$32)^2) +9*$C$33/SQRT((B$1-$A$33)^2+($A2-$B$33)^2) +9*$C$34/SQRT((B$1-$A$34)^2+($A2-$B$34)^2) = = = = 5. Click and drag the corner of B2 to extend this information into your entire chart. = = = = 6. Use this spreadsheet to make a graph. Select a 3D surface chart. Next select a contour surface chart. The maximum should be 100 and the major units should be 5. = = = = 7. Describe the current graph. What does it tell you about the potential outside a uniformly charged sphere? (Repeat this step after each new set of data provided by step 8) = = = = 8. When you have finished recording your initial results, go through each of the following four sets of data and enter them into your spreadsheet. After each one, record your results as you did in step 7. = = = = = = =
 * CATEGORY ||  SCORE (1-4)  ||  POINTS (0-20)  ||
 * Content || 3 || 15 ||
 * Organization || 4 || 20 ||
 * Accuracy || 4 || 20 ||
 * Appearance || 4 || 2 ||
 * Participation || 3.5 || 17.5 ||
 * TOTAL || 92.5 ||