Problem 29 - Physics

Question

Two charges, \(-16\) and \(+4.0 \mu \mathrm{C}\), are fixed in place and separated by \(3.0 \mathrm{~m}\). (a) At what spot along a line through the charges is the net electric field zero? Locate this spot relative to the positive charge. (Hint: The spot does not necessarily lie between the two charges.) (b) What would be the force on a charge of \(+14 \mu \mathrm{C}\) placed at this spot?

Step-by-Step Solution

Answer

The zero field is 3 m to the right of the positive charge, and the force on 14 μC is 0.

Step 1: Define the Problem

We need to find the point along the line joining two charges where the electric field is zero. The charges are \(-16\, \mu \mathrm{C}\) and \(+4\, \mu \mathrm{C}\) separated by \(3.0\, \mathrm{m}\). The zero electric field can be located to the left of the negative charge or to the right of the positive charge.

Step 2: Use the Electric Field Formula

The electric field due to a point charge is given by \(E = \frac{k |Q|}{r^2}\), where \(k = 8.99 \times 10^9\, \mathrm{N\cdot m^2/C^2}\), \(Q\) is the charge, and \(r\) is the distance from the charge. We will calculate where the fields due to both charges cancel each other out.

Step 3: Set Up the Equation

Let \(x\) be the distance from the \(+4\, \mu \mathrm{C}\) charge where the electric field is zero. The electric field at this point due to both charges must cancel each other out. Since the point cannot be between the charges, assume it is \(x\) to the right of the \(+4\, \mu \mathrm{C}\) charge. Thus, \(r_1 = x\) and \(r_2 = x + 3\) for the \(-16\, \mu \mathrm{C}\) charge. The condition for zero electric field is:\[\frac{k \times 4}{x^2} = \frac{k \times 16}{(x+3)^2}\]

Step 4: Solve the Equation

Simplify the equation:\[\frac{4}{x^2} = \frac{16}{(x+3)^2}\]Cross-multiply to get:\[4(x+3)^2 = 16x^2\]Divide the whole equation by 4:\[(x+3)^2 = 4x^2\]Take the square root of both sides:\[x+3 = 2x\]Solve for \(x\):\[x = 3\, \text{m}\]Therefore, the point where the electric field is zero is \(3\, \text{m}\) to the right of the \(+4\, \mu \mathrm{C}\) charge.

Step 5: Find the Force on a New Charge

The force \(F\) on a charge \(q = +14\, \mu \mathrm{C}\) placed at the zero electric field point is given by \(F = E \cdot q\). Since the electric field \(E\) is zero at this point, the force \(F\) is also zero.

Key Concepts

Coulomb's Law

When studying electric fields and forces between charged objects, Coulomb's Law is a fundamental principle to understand. It states that the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This can be mathematically expressed as:\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]where:

• \(F\) is the magnitude of the force between the charges, measured in Newtons.
• \(q_1\) and \(q_2\) are the values of the two charges in Coulombs.
• \(r\) is the distance between the charges in meters.
• \(k\) is Coulomb’s constant, approximately equal to \(8.99 \times 10^9 \text{ N m}^2/ \text{C}^2\).

This law helps explain how charges interact with each other over a distance, creating either an attractive or repulsive force, depending on the sign of the charges.

Point Charges

In the realm of electrostatics, point charges are idealized models of charged objects. They represent locations in space where charge is concentrated at a single point with zero volume. This simplification allows us to easily calculate the electric forces and fields in a system. Point charges are useful because:

• They simplify complex calculations by focusing on the position and magnitude of the charge rather than its shape or volume.
• They help in understanding interactions and fields in environments with simple geometry.
• They are used to derive and apply fundamental laws like Coulomb's Law and the principle of superposition.

Although no physical object is a true point charge, many can be approximated as such, especially when their dimensions are small compared to the distance over which they interact.

Electric Force

Electric force is the interaction between charged objects and is a component of electromagnetic force, one of the four fundamental forces in nature. This force can be attractive or repulsive:

• Like charges (both positive or both negative) repel each other.
• Opposite charges (one positive, one negative) attract each other.

The strength of this force is described by the electric field generated by a charge, and it acts along the line joining the charges. The magnitude is calculated using Coulomb’s Law, providing insight into how charged bodies influence each other from a distance. Understanding this force is crucial in fields such as chemistry, where it dictates atomic and molecular interactions.

Superposition Principle

The superposition principle is pivotal when calculating forces and fields involving multiple charges. It states that the total electric field or force experienced at a point is the vector sum of the fields or forces due to individual charges. This principle simplifies complex problems by allowing:

• Individual computation of fields from each separate charge.
• The combination of these individual effects to find the total field or force.

For instance, if you need to calculate the electric field at a point due to several charges, you calculate the electric field created by each charge at that point and then add them up, taking both magnitude and direction into account. This is particularly helpful in systems with multiple charges where direct computation would be too complex.