If two electrons are each 1.50 ×10^-10 m from a proton (Fig. E21.45), find the magnitude and dir

If two electrons are each 1.50 ×10^-10 m from a proton (Fig....

Key Concepts

Coulomb's Law

Coulomb's Law expresses the force between two point charges, stating that the magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law is a cornerstone in electrostatics and provides the mathematical basis for calculating electric forces between charged particles.

Principle of Superposition

The Principle of Superposition is a fundamental concept in electrostatics which states that the net electric force exerted on a charge by a collection of other charges is the vector sum of the individual forces exerted by each of the charges. This allows for the independent calculation of forces due to each charge and then combining them to find the total force.

Properties of Electric Charge

The behavior of electric forces is largely determined by the properties of electric charge, including the fact that charges can be either positive or negative. Like charges repel while opposite charges attract. Understanding these properties is essential for determining the direction of forces between charged particles.

Vector Addition

Since electric forces have both magnitude and direction, they are treated as vector quantities. Calculating the net force involves vector addition, taking into account the directions of individual forces. This is particularly important when the charges are arranged in a spatial configuration where symmetry or differing distances can influence the resultant force direction.

Transcript

00:01 So here i have already drawn that diagram and mentioned all the variables.

00:06 Now you can see that since q2 and q3 are at this are situated at the same distance from q1 which is a proton this means that there is some symmetry involved.

00:22 We just need to use the correct axis in order to utilize that symmetry.

00:27 So here let's take, since these two distance are equal, we can choose our x -axis to be through the two vectors joining q -1 with q2 and q1 with q3.

00:47 And this means that these two angles will be equal to theta over 2 and theta over 2.

00:54 So let's choose this to be our x -axis, which means that the orthogynes, vector will be equal to the y -axis will be our y -axis positive y -axis and this will be the negative y -axis now as you can see since they have the same charge q2 q3 are both electrons they have the same distance from q1 and they're situated at the same angle from x so this means that you can take advantage of symmetry and you'll see that the y components of each of the force will cancel because they'll have the same magnitudes but opposite directions.

01:48 So this means we can just add the x components that also happen to be equal.

01:57 So the net force will be equal to double of each of any of the x component.

02:02 So let me just write down the equation.

02:05 To show you what i mean.

02:09 So this is the proton.

02:11 Q1 is the proton and we need to find the forces due to q2 and q3 at this point where we have the proton.

02:21 So the force obviously will have two components.

02:24 We can resolve the force, each of the forces into two components.

02:30 So we always know that charge here opposite the force between opposite charges is attractive so force on q1 due to q2 will be towards q2 so like this and similarly force on q1 due to q3 will be towards q3 because both of these forces are attractive so let's call this as f2 meaning that this force is due to q2 and let's call this as f3 which means that this is the force due to q3.

03:16 Now each of these forces will have two components.

03:21 X component.

03:23 Sorry, this is a y component and this will be the x component.

03:28 Similarly, this will be the y component and again there will be another x component over here.

03:38 Me just choose another color for this falls.

03:46 So this is f3 and this is the y component of f3 and this is the x component of f3.

03:56 So let's call the y components to be f2y and this will therefore be f3y and similarly i won't be writing it over here because that will make the diagram really messy.

04:11 So the x component will be like f2x.

04:14 For f2 and f3 x for f3 now let's write down what f2 is just the forces f2 and f3 so f2 will be equal to k k k u1 q2 over r square and i'm not uh writing the direction uh i'm just writing the magnitude here and f3 will similarly be similarly be equal to k q1 q3 over r square.

04:55 Now let's substitute the values, not the algebra values, just the numeric values.

05:03 So q1 and q2 are opposite.

05:07 So q2 is negative of q1.

05:10 This means that this force can be written as to be equal to negative k q1 square over r square.

05:19 And similarly, f3 will also be equal to negative kq1.

05:23 Square over r square so we see that both of these forces have the same magnitudes now let's write on the components you can directly ignore the white component if you get the idea that both of these will be equal and opposite hence they will cancel out but if you don't you can go step by step as to calculate in the components and adding them up so f2y will be equal to negative kq1 square, r square, and this points upward since f2 is in the first quadrant.

06:05 So this will be positive jkap.

06:16 Now one other thing that i should mention over here is we don't need to take this, consider this minus sign over here.

06:30 Since we have already considered the directions as to how they will point.

06:36 So we can just ignore this direction and just substitute the absolute magnitudes of the charges...

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