Problem 24 - University Physics with Modern Physics

Question

For each of the following arrangements of two point charges, find all the points along the line passing through both charges for which the electric potential \(V\) is zero (take \(V = 0\) infinitely far from the charges) and for which the electric field \(E\) is zero: (a) charges \(+Q\) and \(+2Q\) separated by a distance \(d\), and (b) charges \(-Q\) and \(+2Q\) separated by a distance \(d\). (c) Are both \(V\) and \(E\) zero at the same places? Explain.

Step-by-Step Solution

Answer

(a) No points for zero potential, one for zero E-field; (b) Potential zero between charges, E-field never zero; (c) Electric field and potential zero at different points.

Step 1: Understanding Electric Potential for Part (a)

We have two charges, \(+Q\) and \(+2Q\), separated by a distance \(d\). The electric potential \(V\) at a point along the line connecting them is given by:\[ V = \frac{kQ}{r_1} + \frac{k(2Q)}{r_2}, \]where \(r_1\) is the distance from \(+Q\) and \(r_2\) is the distance from \(+2Q\). We need \(V = 0\). Set the equation to zero and solve:\[ \frac{1}{r_1} + \frac{2}{r_2} = 0. \]For the algebra, there are no real solutions along the line between the charges or on the line extending past \(+2Q\), as both terms are positive.

Step 2: Finding Points Along the Line for Electric Field Zero, Part (a)

The electric field \(E\) at a point is found by:\[ E = \frac{kQ}{r_1^2} - \frac{k(2Q)}{r_2^2}, \]as the electric fields due to each charge have opposite directions along the line. Here, we calculate:\[ \frac{1}{r_1^2} = \frac{2}{r_2^2}. \]Solving for \(x\leq 0\) (on the extension past \(+Q\)) gives:\[ r_1 = \sqrt{2} r_2. \]Finding \(x\) with this condition relative to the positions defines one real solution when \(x < 0\), as the fields' magnitudes balance at the ratio of distances.

Step 3: Calculating Electric Potential for Part (b)

Now using charges \(-Q\) and \(+2Q\), the expression for \(V\) becomes:\[ V = \frac{-kQ}{r_1} + \frac{k(2Q)}{r_2}. \]Set the potential to zero and solve:\[ \frac{-1}{r_1} + \frac{2}{r_2} = 0. \]This has a solution not only between the charges but not at infinite distance due to charge configuration, as calculations confirm possible sites between the charges.

Step 4: Finding Electric Field Zero for Part (b)

Find \(E\), we have:\[ E = \frac{-kQ}{r_1^2} - \frac{k(2Q)}{r_2^2}. \]Solve:\[ \frac{-1}{r_1^2} = \frac{2}{r_2^2}, \]imply no real point exists where the balances equalizes, since the fields could not cancel each other—all are directional opposites at each finite point.

Step 5: Answer if Both are Zero at the Same Places Part (c)

Review both (a) and (b):(a) \(V\) not zero at any real point, but \(E\) balanced at one point left of \(+Q\). (b) \(V\) zero point found between, \(E\) isn't zero in finite proximity toward \(-Q\) from \(+2Q\).Electric field and potential generally aren't zero at the same spots without a dipole null component.

Key Concepts

Point Charges

Point charges are fundamental elements in the study of electric fields and potentials.
They are idealized charges that are considered to be located at a single point in space. This simplification allows us to ignore the details of their shape or size while focusing on how they interact with other charges.

• Imagine two point charges, which we'll designate as Charge "+Q" and Charge "+2Q". These charges exert forces and potentials around them.
• The basic principle is that like charges repel each other and opposite charges attract, influencing the potential and electric field in their vicinity.
• The force between point charges follows Coulomb's Law, which states that the force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them.

Understanding how point charges interact is key to analyzing scenarios involving distances and fields, like in this exercise.

Electric Potential Zero

Electric potential arises from the work needed to move a charge within an electric field.
It is similar to the concept of gravitational potential energy. The point where electric potential equals zero is significant since it signifies a balance or neutralization effect of charges involved.

• In our scenario, we consider two cases: the charges "+Q" and "+2Q", and the charges "-Q" and "+2Q".
• For like-charges "+Q" and "+2Q", the electric potential cannot be zero at any point along the line connecting them because these charges create only positive potentials, which cannot cancel out completely.
• When considering "+2Q" and "-Q", the potential can be zero at some point in between them. The positive and negative potentials can actually cancel each other out if the geometry allows it.

The concept of zero electric potential helps us map regions where no net energy is required to position a small test charge.

Electric Field Zero

An electric field represents the force experienced by a unit positive charge. The electric field can be zero at a point if the forces due to different charges cancel each other out.

• In mechanics, a zero electric field resembles equilibrium where forces on a charge are balanced.
• For charges "+Q" and "+2Q", the field can be zero to the left of "+Q", where the field from each charge balances in opposite directions.
• With charges "-Q" and "+2Q", it's essential to note that the fields add rather than cancel out since they are in opposite directions with respect to their geometric placement.

An electric field being zero indicates where a test charge would experience no net force.

Charge Interaction

Understanding charge interaction helps to predict how charges will affect one another.
The interaction not only depends on the amount of charge but also on their spatial arrangement.

• Charges create potential and fields that influence surrounding charges.
• For two like charges "+Q" and "+2Q", there is a repulsion, making it difficult to find points where their potential can become zero.
• The combination of "-Q" and "+2Q" introduces an attraction in the interaction, allowing for a potential zero somewhere along the direct line between them.

It's important to understand how interaction influences the solutions for zero potential and field, as not all configurations offer zeros at the same locations.