Two point charges qA=-12.0 μC and qB=45.0 μC and a third particle with unknown charge qC are loc

Two point charges qA=-12.0 μC and qB=45.0 μC and a third...

Two point charges $q_{\mathrm{A}}=-12.0 \mu \mathrm{C}$ and $q_{\mathrm{B}}=45.0 \mu \mathrm{C}$ and a third particle with unknown charge $q_{\mathrm{C}}$ are located on the $x$ axis. The particle $q_{\mathrm{A}}$ is at the origin, and $q_{\mathrm{B}}$ is at $x=$ $15.0 \mathrm{cm} .$ The third particle is to be placed so that each particle is in equilibrium under the action of the electric forces exerted by the other two particles. (a) Is this situation possible? If so, is it possible in more than one way? Explain. Find (b) the required location and (c) the magnitude and the sign of the charge of the third particle.

Two point charges $q_{\mathrm{A}}=-12.0 \mu \mathrm{C}$ and $q_{\mathrm{B}}=45.0 \mu \mathrm{C}$ and a third particle with unknown charge $q_{\mathrm{C}}$ are located on the $x$ axis. The particle $q_{\mathrm{A}}$ is at the origin, and $q_{\mathrm{B}}$ is at $x=$ $15.0 \mathrm{cm} .$ The third particle is to be placed so that each particle is in equilibrium under the action of the electric forces exerted by the other two particles. (a) Is this situation possible? If so, is it possible in
more than one way? Explain. Find (b) the required location and (c) the magnitude and the sign of the charge of the third particle.

Key Concepts

Superposition Principle

The superposition principle states that when multiple sources of electric force are present, the total force experienced by a charge is the vector sum of the forces exerted by each charge independently. This concept is crucial for analyzing systems with more than two charges, as it allows for the independent calculation of forces.

Coulomb's Law

This law states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. It provides the fundamental mathematical framework for calculating the magnitude and direction of the electric force acting between charges.

Electric Force Equilibrium

Electric force equilibrium occurs when the net force acting on a charge due to other charges is zero. This requires that the vector sum of all forces acting on the charge cancels out, which is essential when determining positions where a charge can be placed so that it experiences no net force.

Effect of Charge Sign on Force Direction

The sign of a charge determines the nature of the force it exerts: like charges repel while opposite charges attract. Understanding the role of charge sign is essential for predicting the direction of forces in arrangements of multiple charges and for determining conditions under which equilibrium can be achieved.

Transcript

00:01 So for the first question, yes, it is possible, but there's only one situation.

00:04 Okay? the only one situation is when the third particle charge is placed at the left of the origin point, which is along the negative x -axis.

00:16 Okay? it's because the magnitude of the charge qa is less than the charge qb.

00:21 So in order to balance out each particle through the equilibrium, the distance between the third charge particle to charge qa, must be less than a distance from third charged particle to charge qb.

00:36 Okay? so therefore, let's jump to question b to determine the distance.

00:46 Well, we know since each particle must be experiencing equilibrium if the third particle was joined.

00:53 Therefore, we have such equation here, which is let's say the electric force due to charge qa has the same magnitude of the electrical, pressure force due to charge qb on a third particle.

01:07 Okay? so therefore we have such equation here, which is k times 12, oh sorry, it times absolute value q a times q a over x square is equal to k times q b times q over x plus 15 square.

01:41 Okay.

01:43 So, so k is k is constant, and ka, qa, qb is the charge qaqb.

01:47 The reason i put absolute value here, because we're talking about the magnitude, and charge qa is negative charge, so i put absolute value.

01:52 And q is the charge on third charge particle, okay? x is the distance from the third charged particle to charge qa, and x plus 15 is the distance from a third charged particle to charge qb.

02:05 And 15 here is the x equal to 15 centimeter away from the origin point, okay? so therefore, as you can tell, q and q can be cancelled, k and k and k can be canceled.

02:20 Therefore, we have absolute value qa over x squared is equal to qb over x plus 15 square.

02:35 Okay? then we can plug in value.

02:38 We have absolute value qa, just simply 12 micropoleon over x square, which is equal to 45 microcolon over x plus 15 square.

02:54 Okay.

02:56 And then we can cancel the units, okay? because eventually we'll put units on our answer later, okay? so then we'll have 12 times x plus 15 square is equal to 45 x square.

03:16 Then we'll have 33x squared minus 360x plus, i'm sorry, minus 2 ,700 is equal to 0.

03:29 And then you can divide by 3 on each side, or you can say times 1 third.

03:35 Then we'll have the 7x2x minus 120x minus 900 equal to 0.

03:46 Then we can apply such formula, negative b plus minus squared, rule b squared minus 4 ac over 2a.

04:00 So we have 120 plus minus square root 120 plus 4 times 11 and times 900 over 2 times 11 and this will give us two solution.

04:28 One is a positive solution which is 16 because remember unit is a centimeter, okay? another one is negative...

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