Problem 17 - University Physics with Modern Physics

Question

Point charges \(q_{1}=+2.00 \mu \mathrm{C}\) and \(q_{2}=-2.00 \mu \mathrm{C}\) are placed at adjacent corners of a square for which the length of each side is 3.00 \(\mathrm{cm} .\) Point \(a\) is at the center of the square, and point \(b\) is at the empty corner closest to \(q_{2}\) . Take the electric potential to be zero at a distance far from both charges. (a) What is the electric potential at point \(a\) due to \(q_{1}\) and \(q_{2} ?\) (b) What is the electric potential at point \(b ?\) (c) A point charge \(q_{3}=-5.00 \mu \mathrm{C}\) moves from point \(a\) to point \(b .\) How much work is done on \(q_{3}\) by the electric forces exerted by \(q_{1}\) and \(q_{2} ?\) Is this work positive or negative?

Step-by-Step Solution

Answer

The electric potential is 0 V at both points; no work is done on the charge.

Step 1: Calculate the distance from charge to center

Since the point \(a\) is at the center of the square, which has a side length of \(3.00\ \text{cm}\), we need to determine the distance of point \(a\) from either \(q_1\) or \(q_2\). Using the Pythagorean theorem, this distance can be calculated as follows:\[ \text{Distance} = \frac{\sqrt{3.00^2 + 3.00^2}}{2} = \frac{\sqrt{18}}{2} = \frac{3\sqrt{2}}{2}\ \text{cm} \]

Step 2: Calculate the electric potential at point a

The electric potential \(V\) at a point due to a point charge \(q\) is given by:\[ V = \frac{k \cdot q}{r} \]where \(k\) is Coulomb's constant \((8.99 \times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2)\), \(q\) is the charge, and \(r\) is the distance from the charge. For the potential at point \(a\) due to \(q_1\) and \(q_2\):\[ V_a = \frac{k \cdot 2.00 \times 10^{-6}}{(3\sqrt{2}/2)\times10^{-2}} + \frac{k \cdot (-2.00 \times 10^{-6})}{(3\sqrt{2}/2)\times10^{-2}} = 0\ V \]

Step 3: Calculate the electric potential at point b due to q1

The potential at point \(b\) due to charge \(q_1\) is:\[ V_{b1} = \frac{k \cdot 2.00 \times 10^{-6}}{3.00 \times 10^{-2}} \]

Step 4: Calculate the electric potential at point b due to q2

The potential at point \(b\) due to charge \(q_2\) is:\[ V_{b2} = \frac{k \cdot (-2.00 \times 10^{-6})}{3.00 \times 10^{-2}} \]

Step 5: Total electric potential at point b

The total electric potential at point \(b\) is the sum of the potentials from \(q_1\) and \(q_2\):\[ V_b = V_{b1} + V_{b2} = \left( \frac{k \cdot 2.00 \times 10^{-6} - k \cdot 2.00 \times 10^{-6}}{3.00 \times 10^{-2}} \right) = 0\ V \]

Step 6: Compute work done moving q3 from a to b

The work done \(W\) by the electric forces when moving charge \(q_3\) from \(a\) to \(b\) can be found by the equation:\[ W = q_3(V_b - V_a) \]Given that both \(V_b\) and \(V_a\) are zero from our previous calculations, the work done is:\[ W = (-5.00 \times 10^{-6})(0 - 0) = 0\ \text{J} \]

Step 7: Determine if the work is positive or negative

Since the work done \(W\) is zero, it is neither positive nor negative. The movement of charge \(q_3\) from \(a\) to \(b\) does not affect the system energetically under the influence of \(q_1\) and \(q_2\).

Key Concepts

Point charge

A point charge is a simplified electrical charge that is considered to be at a single, infinitely small point in space. This concept is crucial in physics as it allows us to simplify complex problems by isolating individual charges rather than dealing with extended charge distributions.
In practical scenarios, point charges are used to model various physical situations where the size of the charged object is negligible compared to other distances involved in the problem, like the distance between different charges or the dimensions of the setup.

• Point charges are often denoted by a letter, typically "q," followed by a subscript to indicate a specific charge (e.g., \(q_1, q_2\)).
• The force field created by a point charge decreases with distance, making distance a crucial factor when calculating effects such as electric potential and electric field.

This brings us to the analysis of electric potentials created by point charges in different locations which is central to understanding how charges interact with each other in space.

Coulomb's law

Coulomb's law is a fundamental principle used to describe the force between two charged objects. It states that the force (F) between two charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. It can be written as:
\[F = k \frac{|q_1 \cdot q_2|}{r^2}\]
where:

• \(k\) is Coulomb’s constant, \(8.99 \times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2\)
• \(q_1\) and \(q_2\) are the point charges
• \(r\) is the distance between the charges

This law not only aids in calculating forces but also electric fields and potentials, as seen in our step-by-step solution.
In the exercise, when calculating the potential at the center and corner of a square, Coulomb's law is essentially being used to justify the calculation of distance factors and why the potential at point \(b\) and point \(a\) results in cancellations of forces.

Electric field

An electric field (E) is a region around a charged particle where forces would be exerted on other charges. It is an essential concept for understanding how charged particles interact over a distance. The electric field due to a point charge \(q\) is given by:
\[E = \frac{k \cdot q}{r^2}\]
where:

• \(E\) is the electric field strength
• \(k\) is Coulomb’s constant
• \(q\) is the point charge generating the field
• \(r\) is the distance from the charge to the point where the field is being measured

An electric field indicates the direction that a positive test charge would move if placed in the field, which means electric field vectors always point away from positive charges and toward negative charges.
Understanding electric fields helps in visualizing potential energy landscapes. It sets the stage for further calculations involving electric potential and potential energy differences, which were integral to the described exercise and are understood using the superposition principle just as the electric potential calculations in this problem.