In a certain region of space, the electric potential is V(x, y, z)=A x y-B x^2+C y, where A, B,

In a certain region of space, the electric potential is V(x,...

Key Concepts

Electric Potential

Electric potential is a scalar field that represents the potential energy per unit charge at a point in space. It allows one to determine how much work would be needed to move a charge from one point to another in an electrostatic field and provides a way to compute the corresponding electric field through spatial derivatives.

Electric Field

The electric field is a vector field that describes the force per unit charge exerted on a test charge placed at any point in space. It is closely related to the electric potential, and its direction is given by the negative gradient of the potential, indicating the direction of maximum decrease.

Gradient

The gradient is an operation applied to a scalar field that produces a vector field representing the rate and direction of the fastest increase of the scalar function. In the context of electrostatics, taking the negative of the gradient of the electric potential yields the electric field.

Critical Points in Electrostatics

Critical points in electrostatics, where the electric field is zero, are found by setting the negative gradient of the electric potential equal to zero. These points indicate locations of equilibrium, where a test charge would experience no net force, and analyzing them often involves solving a system of equations derived from the partial derivatives of the potential.

Transcript

00:01 All right, so to begin, we must identify that the electric field is a negative gradient of the potential, and it's mathematically shown by this following equation.

00:13 So the electric field is the derivative of so dv over d r.

00:26 So now given that the potential, so now if he assumed that the potential is v, let me, let me use, let me use, different colors so if you seem that the potential is b so in the positions so if we computed in the into cartesian form this is the c so now if you say that the potential at a given point is this so a x y minus b x squared plus c y so here a b c or constants so now we can execute that the x component the field is, so you can combine these two, we switch back to black, so it's the electric field at point x is equal to, once again, it's equal to this.

01:25 And now we can compute this as the, so the radius is dx.

01:32 So we can change the r to dx.

01:35 And now the potential is just this, let me copy it, move it right here.

01:49 Yeah, so as i said before, now if you move it there, yeah.

01:57 So now if we execute this and just derive this with respect to x, you can get that it's equal to negative, a .y plus 2bx.

02:11 Since when we derive it, just the 2 goes there and the x becomes of singular power.

02:20 And since there is no x here, the constant just goes to 0.

02:25 So now, not to find the y component of the field.

02:30 So if you did the y component, so at y, now once again, we're going to have to derive it, so with dy.

02:39 So if you copy this again, and now we derive it.

02:52 So we're going to get that.

02:56 So now if you derive it with respect to y, since there's y to the power 1 here, it just becomes a x minus x.

03:05 Don't forget the minus in front of the d here.

03:08 Minus and since there is no by constant here this just turns to zero and this there is y to the power 1 so it becomes coefficient which is just c so now if you find the electric potential at c so d over d c and as you can already probably tell just by looking at the sorry so this dz hold on let me let me write this a little bit better so this is the c.

03:51 So as you can probably tell just by looking at the green formula here, there is no z constant, so we can already guess that the, here, let me just move it first.

04:03 So we can know for sure.

04:07 Get this whole thing over down here.

04:11 So now since there is no z constant here, this will just equal zero.

04:17 So the potential at c is equal to zero.

04:21 And that's the answer to the first part.

04:24 And now, for part b, for part p, for the electric field to be zero, each component of the electric field must be zero.

04:37 So that means all these components must be zero...