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Example 1- Potential of a point charge
Now let’s calculate the potential of a point charge. Let’s assume that we have a positive point charge, q, sitting over here, and now we know that it generates electric field in radially outward direction, filling the whole space surrounding the charge and going from charge to the infinity in radially out direction. The potential, by choosing the 0 potential at infinity, was defined as minus integral of E dot dr, integrated from infinity to the point of interest in space.
Let’s assume that the point that we’re interested is over here and it is r distance away from the source. dr is the incremental displacement vector in radial direction and recall that electric field is q over 4π ε0 r2 for a point charge. It is of course radially outward direction for a positive charge. Then the potential of this charge becomes equal to minus magnitude of the first vector, and that is q over 4π ε0 r2, magnitude of the second vector dr, and again, dr is an incremental displacement vector in radial direction. An electric field is also in radial direction at this point. Therefore the angle between these two vectors is 0 degrees, so we have here then cosine of 0 as a result of this dot product.
This quantity will be integrated from infinity to the point of interest, which is located r distance away from the charge. Cosine of 0 is 1 and q over 4π ε0 constant can be taken outside of the integral and potential V, therefore becomes equal to –q over 4π ε0 times integral of dr over r2 integrated from infinity to r. Integral of dr over r2 is -1 over r, so V is equal to minus q over 4π ε0 times -1 over r evaluated at infinity and r.
This minus and that minus will make a positive and if you substitute r for the little r in the denominator we will have q over 4π ε0 r. If we substitute infinity for little r, then the quantity will go to 0 because any number divided by infinity goes to 0. Therefore our result is going to be that the potential of a point charge is equal to charge divided by 4π ε0 times r.
Here r is the distance between the point charge and the point of interest. Here we should also make an important note, as you recall that the potential was electrical potential energy U per unit charge. Here, energy is a scalar quantity, charge is also a scalar quantity, and whenever we divide any scalar by a scalar, we end up also with a scalar quantity. Therefore potential does not have any directional properties. It is not a vector, and that makes also dealing with potential much easier than dealing with the electric field, because we don’t have to worry about any directional properties for this case.
If we have more than one point charge in our region of interest, then since we are dealing with scalar quantities, we can calculate the potential of the specific point simply by calculating the potentials generated by each individual point charges at the location of interest and then simply adding them. If we make a note of that over here, for more than one point charge, for example if I have q1 and q2 and q3 and so on and so forth, and if I am interested with the potential at this point, I look at the distances of these charges to the point of interest and calculate their potentials. V1 will be equal to q1 over 4π ε0 r1 over and let’s give some sign to these charges also.
Let’s say that this is positive, this is negative, this is positive. V1 will be q1 over 4π ε0 r1. V2 is going to be equal to, again this is positive charge q2 over 4π ε0 r2, and V3 is going to be equal to, since it is negative, –q3 over 4π ε0 r3. Once we determine their potentials relative to this point, then the total potential will be equal to V1 plus V2 plus V3, or is going to simply be equal to 1 over 4π ε0 will be common, q1 over r1 plus q2 over r2 minus q3 over r3. Therefore the total potential that this system of charges generates at this point P is going to be equal to this quantity.