(a) Find the electric field at the center of the triangular configuration of charges in Figure 1

step 1 Welcome back to Numeran. My name is Kevin Chirac. Let's consider here that we have an equilatera

(a) Find the electric field at the center of the triangular...

Key Concepts

Vector Addition

Since electric fields are vector quantities, both their magnitudes and directions must be considered when combining them. Vector addition is used to determine the resultant electric field at a point by taking into account the individual field components contributed by each charge.

Superposition Principle

The superposition principle states that the total electric field produced by multiple charges is the vector sum of the electric fields due to each individual charge. This concept is essential for solving problems involving multiple sources, such as the triangular configuration, as it allows one to calculate the net effect by considering each charge’s contribution separately.

Equilibrium Conditions

Equilibrium in the context of electric fields occurs when the net electric field at a point becomes zero due to the cancellation of the vectors contributed by the surrounding charges. Understanding the conditions required for equilibrium is important for exploring scenarios, such as finding if other combinations of charges can produce a net zero electric field at the center of a configuration.

Electric Field

The electric field is a vector field that represents the force per unit charge exerted on a test charge at any point in space. It is defined through Coulomb's law for point charges and plays a key role in understanding how charges interact through forces that vary inversely with the square of the distance between them.

Coulomb’s Law

Coulomb's law provides the quantitative measure to calculate the force between two point charges. This law specifies that the magnitude of the force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them, which is fundamental when determining the electric field contribution of individual charges.

Transcript

00:01 Welcome back to numeran.

00:02 My name is kevin chirac.

00:03 Let's consider here that we have an equilateral triangle.

00:06 And in this equilateral triangle, we have three different charges.

00:10 And we're going to be looking for how these charges affect the electric field at the center.

00:14 So we have charge a, b, and c.

00:19 Now we know that if we're going to be looking at the effect that it has in the center, but we're going to need to be considering, because this is the vector system, two different things.

00:25 The first is going to be the magnitude of the sum of all the electric fields, which is going to be equal to the effect.

00:32 The square root of the sum of all the electric fields that are present and just their x component, squaring that, and then adding that to the sum of all electric fields y component and squaring that.

00:47 So it's going to be a nice little pythagorean theorem here that we're going to be using.

00:50 And then we'll eventually need to figure out what the angle of direction is as well.

00:55 So let's begin with figuring out what the sum of all of the electric fields are in the x direction.

01:00 We know by now that we can recognize this to be a system where, the k constant is going to be constant in every single one of these terms that we're about to add up.

01:09 Similarly, the radius is going to be similar, or identical in both of these.

01:17 So the k and the r squared are going to be the same.

01:20 And really all we need to be considering is, for example, qa times the scale required to get just the x component of a.

01:31 So if we take a look over here, charge a is going to.

01:35 To be positive and thus emitting electric field lines away from the charge so its effect on this center dot is actually going to be straight down so if we're looking for the x component that would be cosine of 90 degrees we would then want to do the exact same thing for the next one that'd be qb qb is going to be pushing to the right and pulling down and that's going to be because it's a negative charge the negative charge is going to be pulling down this way and then for qc it's going to be pushing up so we can see that all of this is going to be qb times for the x component it would be the cosine of 30 degrees and then plus qc also cosine 30 this is going to simplify down to k over r squared where we then have qa or the magnitude of qa times zero plus qb times one half plus qc times one half.

02:51 We know that the constant is 8 .9 times 10 to the 9th, r, we can do a little calculation over here.

02:59 Since this entire length here is 0 .25, we can divide it in half, and then we can use that divided by the cosine of the angle here, which is 30, to find out what this hypotenuse is.

03:12 So all of that to say that the radius is going to be a 0 .25, divided by 2, divided by cosine 30, which comes out to be, 0 .144.

03:30 Good so we plug that in over here now we have 0 .144 squared we know the magnitudes of all these charges so that's going to be 8 times 10 to the negative 9th or qb times 1 half plus 1 times 10 to the negative 9th times root 3 over 2 or excuse me times times 1 half for qc excuse me this should be root 3 over 2 if we're looking here at the component that is in the x direction with something running off this at the way, that angle here is going to be 30 degrees.

04:21 And so we've been looking for cosine of 30 degrees, which would be root 3 or 2, so let me correct that real quick.

04:30 R3 or 2 and root 3 over 2.

04:32 Good...

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