Question

A charge of \(-1 \mathrm{nC}\) is located at the origin in free space. What charge must be located at \((2,0,0)\) to cause \(E_{x}\) to be zero at \((3,1,1)\) ?

Short answer

The required charge at point (2,0,0) is approximately -0.61 nC.

Step-by-step solution

Understand the problem and given information

We are given a charge of -1nC at the origin in free space. We need to find the charge at point (2,0,0) such that E_x at the point (3,1,1) becomes zero. Let the charge at point (2,0,0) be \(q_2\).

Calculate distance between each charge and point (3,1,1)

First, calculate the distance between each charge and the point (3,1,1). For the charge at the origin: \(r_1 = \sqrt{(3-0)^2+(1-0)^2+(1-0)^2}=\sqrt{9+1+1}= \sqrt{11}\) For the charge at point (2,0,0): \(r_2=\sqrt{(3-2)^2+(1-0)^2+(1-0)^2}=\sqrt{1+1+1}= \sqrt{3}\)

Calculate the electric field components due to each charge

Now, we need to find the electric field components at point (3,1,1) due to both charges. For the charge at the origin: \(E_{1x} = \frac{kq_1 (x_1-x)}{(r_1)^3} = \frac{k(-1 \times 10^{-9})(3-0)}{(\sqrt{11})^3}\) \(E_{1y} = \frac{kq_1 (y_1-y)}{(r_1)^3} = \frac{k(-1 \times 10^{-9})(1-0)}{(\sqrt{11})^3}\) \(E_{1z} = \frac{kq_1 (z_1-z)}{(r_1)^3} = \frac{k(-1 \times 10^{-9})(1-0)}{(\sqrt{11})^3}\) For the charge at point (2,0,0), let's not calculate the actual values of \(E_{2x}\), \(E_{2y}\) and \(E_{2z}\), but set up the expressions for them as they depend on q2, which we are trying to find. \(E_{2x} = \frac{kq_2 (x_2-x)}{(r_2)^3} = \frac{kq_2 (3-2)}{(\sqrt{3})^3}\) \(E_{2y} = \frac{kq_2 (y_2-y)}{(r_2)^3} = \frac{kq_2 (1-0)}{(\sqrt{3})^3}\) \(E_{2z} = \frac{kq_2 (z_2-z)}{(r_2)^3} = \frac{kq_2 (1-0)}{(\sqrt{3})^3}\)

Set up the equation for the x-component of the total electric field to be zero and solve for q2.

Since the x-component of the total electric field at point (3,1,1) needs to be zero, we can write: \(E_{1x}+E_{2x}=0\) \(\frac{k(-1 \times 10^{-9})(3-0)}{(\sqrt{11})^3}+\frac{kq_2 (3-2)}{(\sqrt{3})^3}=0\) Now, simplify the equation and solve for q2: \(q_2=\frac{-1 \times 10^{-9}(\sqrt{3})^3}{(\sqrt{11})^3}\) \(q_2 \approx -0.61 \times 10^{-9} C\)

Final answer

To cause E_x to be zero at point (3,1,1), the charge q2 must be located at (2,0,0) with an approximate charge of \(-0.61 \mathrm{nC}\).

Key concepts

Electric Field Components

An electric field is a vector field around charged particles. It describes how the electric force is distributed in space. Electric fields have three components, usually expressed in terms of their x, y, and z directions. Each component describes the field's impact along one particular axis. This helps break down the complex phenomena of electromagnetic interactions into more manageable parts.

When calculating the electric field at any point due to a single charge, we use the expression:

• \[ E_x = \frac{kq (x_1-x)}{(r)^3} \]
• \[ E_y = \frac{kq (y_1-y)}{(r)^3} \]
• \[ E_z = \frac{kq (z_1-z)}{(r)^3} \]

Where \( E_x, E_y, E_z \) are the electric field components and \(k\) is Coulomb’s constant. The distances \((x_1-x), (y_1-y), (z_1-z)\) denote the difference in coordinates between the charge point and observation point, while \(r\) is the distance between them.

In our problem, we calculated these components to ensure that they interact in such a way that the net electric field in the x-direction was zero at a specific point.

Coulomb's Law

Coulomb's Law is fundamental in electrostatics. It quantifies the amount of force between two stationary, electrically charged particles. The law states:

• The magnitude of the force between two charges is directly proportional to the product of the absolute values of the charges.
• It's inversely proportional to the square of the distance between their centers.

The formula is represented as: \[ F = k \frac{|q_1 q_2|}{r^2} \] where \(F\) is the force between charges \(q_1\) and \(q_2\), \(r\) is the distance between charges, and \(k\) is Coulomb’s constant, approximately \(8.99 \, \times \, 10^9 \) N m extsuperscript{2} C extsuperscript{-2}.

In the given exercise, we applied Coulomb's law indirectly to find the electric field caused by each charge. By setting the x-component of these fields equal to zero and solving for unknown charges, we ensured that the force components effectively canceled each other out in the desired direction.

Charge Distribution

Charge distribution is about how electric charge is disbursed in space. Usually, charges are distributed in various configurations, like point charges, line charges, or surface charges. Each configuration influences how the electric field behaves around them.

In this problem, we dealt specifically with point charges located at specific coordinates. Understanding how to manage charge distribution is crucial for determining fields and forces in practical scenarios.

A point charge is a hypothetical charge located at a single point. When determining the effect of charge distribution, especially for point charges, we calculate the resultant electric field at various points in space. This involves considering both the magnitude and direction of electric fields generated by each individual charge. These fields are then summed to find the overall effect at a given point.

Problem-Solving in Physics

Solving physics problems effectively often involves a strategic approach that includes breaking the problem into smaller, more manageable parts. The original exercise illustrates several key strategies that are crucial in problem-solving:

• **Understand the Problem:** Read the question thoroughly and identify the given information and what needs to be calculated.
• **Visualize the Scenario:** Drawing a diagram can help understand the spatial relationship between charges and points.
• **Apply Relevant Laws and Formulas:** Use laws such as Coulomb's Law or formulas for electric field components as needed.
• **Solve Systematically:** Calculate step-by-step while keeping track of each quantity and operation, making it easy to find where a mistake may have occurred.
• **Check for Reasonableness:** After calculating, consider if the solution makes sense in the context of the problem's physical setup.

By practicing this structured approach to physics problems, students can gain deeper insights not only into solving particular problems but understanding the underlying principles better.