(a) Three charges are situated at the corners of a square...

(a) Three charges are situated at the corners of a square (side $a$ ), as shown in Fig. 2.41. How much work does it take to bring in another charge, $+q$, from far away and place it in the fourth corner? (b) How much work does it take to assemble the whole configuration of four charges?

(a) Three charges are situated at the corners of a square (side $a$ ), as shown in Fig. 2.41. How much work does it take to bring in another charge, $+q$, from far away and place it in the fourth corner?
(b) How much work does it take to assemble the whole configuration of four charges?

Key Concepts

Electrostatic Potential Energy

This concept refers to the work required to assemble a system of point charges from infinity to a given configuration. It’s calculated by summing the potential energy contributions from every pair of charges, where each contribution is determined using Coulomb’s law. Understanding this concept is key to analyzing how energy is stored in systems with multiple charges.

Superposition Principle

The superposition principle in electrostatics states that the total electric potential at any point is the sum of the potentials due to the individual charges. This allows us to calculate complex configurations by analyzing one charge at a time, and then adding up all the contributions to determine the net effect.

Work Done in Assembling Charge Configurations

This key concept involves calculating the external work performed to bring charges together from infinity to a specific layout. This work is equal to the electrostatic potential energy of the system, taking into account the interactions between charges introduced sequentially in the assembly process.

Pairwise Interaction

In systems of multiple charges, the interaction energy between any two charges contributes to the overall electrostatic potential energy. By calculating the energy between each unique pair of charges using Coulomb’s law and then summing these contributions, one can determine the total energy required to assemble the configuration.

Transcript

00:01 Hi there, so for this problem, we have three charges that are situated at the corners of a square of side a, as is shown in this figure.

00:11 Now, the question in this is how much work does take to bring in another charge positive q from far away and place it in the fourth corner in this one right here? so to solve this, we start with the potential that we know is going to be 1 over 4 times pi times epsilon sub 0 times the sum of the charges divided by the position of those charges.

00:49 In this case, we know that there is going to be a charge at a position a negative charge.

01:00 Another charge at a position a as well, and another charge, the one that is in the opposite corner.

01:11 And to obtain that distance, we use the pythagarin theorem, and we will obtain that the length in this is going to be the square root of 2 times a.

01:24 So with that, we can write that the potential is equal to 1 over 4 times pi times epsilon sub 0, minus q divided by a plus q divided by the square root of 2 times a plus minus q divided by a.

01:58 So with this, we can simply simplify this as the charge q divided by 4 times pi times epsilon x0 times a times minus 2 plus 1 over the square root of 2.

02:18 So we know that the word done on the fourth charge is equal to the charge q times the potential.

02:30 So we just, we're going to have that this is just simply the charge q square divided by four times pi times epsilon sub zero times a.

02:41 And this times minus 2 plus 1 over the square root of 2.

02:49 So that's a solution for part a of this problem.

02:55 Now for part b, we are asked about how much work does it take to assemble the whole configuration of four charges? so we start with the work that we have done to put the first charge, charge 1.

03:15 We know that in this case, there is no other charges present at that moment...

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