Question

Three charges, \(q_{1}, q_{2},\) and \(q_{3},\) are located at the corners of an equilateral triangle with side length of \(1.2 \mathrm{~m}\). Find the work done in each of the following cases: a) to bring the first particle, \(q_{1}=1.0 \mathrm{pC},\) to \(P\) from infinity b) to bring the second particle, \(q_{2}=2.0 \mathrm{pC}\) to \(Q\) from infinity c) to bring the last particle, \(q_{3}=3.0 \mathrm{pC}\) to \(R\) from infinity d) Find the total potential energy stored in the final configuration of \(q_{1}, q_{2},\) and \(q_{3}\)

Short answer

Answer: The total potential energy stored in the final configuration of the three charged particles is \(8.25 \times 10^{-11} J\).

Step-by-step solution

Calculate the work done in case a

In this case, we are only concerned with the work done to bring particle \(q_1 = 1.0 \ pC\) to point \(P\) from infinity. The force on particle \(q_1\) is zero while it is at infinity, so no work is required to bring it to \(P\). Therefore, the work done is \(0 \ J\).

Calculate the work done in case b

Now, we will find the work done to bring the second particle, \(q_2=2.0 \ pC\) to point \(Q\) from infinity. The work is equal to the change in electric potential energy between the initial position (at infinity) and the final position (\(Q\)). The electric potential energy between two point charges is given by the formula \(W = \frac{k \cdot q_1 \cdot q_2}{r}\), where \(r\) is the distance between the charges and \(k\) is the Coulomb's constant, \(9 \times 10^9 Nm^2/C^2\). The distance between \(q_1\) and \(q_2\) is equal to the side of the equilateral triangle, \(1.2 \ m\). Plugging the values into the formula, we find the work done in case b: \(W_b = \frac{(9 \times 10^9 Nm^2/C^2)(1.0 \times 10^{-12}C)(2.0 \times 10^{-12}C)}{1.2m} = 1.5 \times 10^{-11} J\)

Calculate the work done in case c

Finally, we'll calculate the work done to bring the last particle, \(q_3=3.0 \ pC\) to point \(R\) from infinity. This time, we have to consider the potential energy, or work done, due to both \(q_1\) and \(q_2\). The total work done will be the sum of the work done in both of these cases. \(W_{c1} = \frac{(9 \times 10^9 Nm^2/C^2)(1.0 \times 10^{-12}C)(3.0 \times 10^{-12}C)}{1.2m} = 2.25 \times 10^{-11} J\) \(W_{c2} = \frac{(9 \times 10^9 Nm^2/C^2)(2.0 \times 10^{-12}C)(3.0 \times 10^{-12}C)}{1.2m} = 4.5 \times 10^{-11} J\) Therefore, the total work done in case c, \(W_c = W_{c1} + W_{c2} = 6.75 \times 10^{-11} J\)

Find the total potential energy stored in the final configuration

Now that we know the work done to bring all the charges to their final positions, we can add them up to find the total potential energy stored in the configuration. \(U_{total} = W_a + W_b + W_c = 0 + 1.5 \times 10^{-11} J + 6.75 \times 10^{-11} J = 8.25 \times 10^{-11} J\) Thus, the total potential energy stored in the final configuration of \(q_1, q_2\) and \(q_3\) is \(8.25 \times 10^{-11}J\).

Key concepts

Work Done by Electric Force

The work done by an electric force is a fundamental concept in electrostatics that describes the energy required to move a charge in the presence of an electric field. Think of it like dragging a heavy object across a room - it takes effort, and that effort is equivalent to work done in physics.

For charges in an electric field, this work is translated into potential energy changes. No work is needed to move a charge from infinity to a point where no other charges are present because the electric force there is zero. However, once other charges are in place, moving an additional charge requires work against the electric forces exerted by the existing charges. This is key to understanding how potential energy builds up in an electrostatic system.

Coulomb's Law

Imagine you're throwing a ball to your friend: the force you apply depends on how hard you throw and how heavy the ball is. In the world of electric charges, Coulomb's Law is the rule that tells us how hard the 'electrical throw' is. It states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

The mathematical formula is given by F = k * (|q1 * q2| / r^2), where F is the force, k is Coulomb’s constant (approximately 9 x 10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges. This fundamental law underpins many principles in electrostatics and is essential for calculating electric forces and potential energy.

Equilateral Triangle Charge Configuration

Placing charges on the corners of an equilateral triangle creates a symmetrical configuration that's a popular choice for demonstrating electrostatic principles. Each charge experiences forces due to the other two charges. The symmetry of an equilateral triangle, where all sides are equal, simplifies the math since the distance between each pair of charges is the same.

In such a triangle, when calculating the potential energy (like in our exercise), each interaction can be considered separately and then summed up for the total energy. It's like assessing the tension in a three-way friendship – each pair contributes to the overall dynamic of the group.

Electric Potential Energy Formula

Electric potential energy is the energy a stationary charge has due to its position in an electric field. To calculate it, you can use the formula U = k * (q1 * q2) / r, similar to Coulomb's law but with a key difference: it's energy, not force. Where U is the electric potential energy, k is Coulomb’s constant, q1 and q2 are the charges, and r is the distance between them.

In the given problem, this formula lets us calculate the work done to bring each charge together in the triangle configuration. The final potential energy of the system is the sum of the work done to assemble the configuration. It's like putting together a puzzle – each piece adds to the final picture, or in this case, the total potential energy.