Question

A positive charge \(q_{1}=1.00 \mu \mathrm{C}\) is fixed at the origin, and a second charge \(q_{2}=-2.00 \mu \mathrm{C}\) is fixed at \(x=10.0 \mathrm{~cm} .\) Where along the \(x\) -axis should a third charge be positioned so that it experiences no force?

Short answer

Answer: The third charge can be positioned either at 3.35 cm or 9.96 cm away from the positive charge along the x-axis to experience no net force.

Step-by-step solution

Identify the charges and their locations

Given, q1 is a positive charge of 1.00 μC, located at the origin (x=0) and q2 is a negative charge of -2.00 μC, located at x=10.0 cm. We need to find the position of a third charge q3 along the x-axis such that it experiences no force.

Consider the force due to positive charge

Let's assume that the third charge (q3) is placed at a distance 'd1' on the right side of positive charge (q1). Since it is placed at the right side, it will experience repulsive electrostatic force due to the positive charge. F1 = k * (q1 * q3) / d1^2 Here, k is the Coulomb's constant, and F1 is the force experienced by q3 due to q1.

Consider the force due to the negative charge

The distance between the negative charge (q2) and the third charge (q3) is d2, where d2 = 10.0 cm - d1. Since the negative charge attracts the positive charges, q3 will experience attractive electrostatic force due to the negative charge q2. F2 = k * (q2 * q3) / d2^2 Here, F2 is the force experienced by q3 due to q2.

Set up the equation for no force condition

The third charge q3 will experience no force if the magnitudes of the forces created by q1 and q2 are equal. Therefore, F1 = F2 k * (q1 * q3) / d1^2 = k * (q2 * q3) / d2^2 For simplicity, let's define r = d1 / d2, then q1 * d2^2 = q2 * d1^2 q1 * (10 - d1)^2 = q2 * d1^2 1 * (10 - d1)^2 = -2 * d1^2

Solve for the unknown distance (d1)

Now, we have to solve the equation to find the value of d1: (10 - d1)^2 = -2 * d1^2 100 - 20 * d1 + d1^2 = -2 * d1^2 3 * d1^2 - 20 * d1 + 100 = 0 This is a quadratic equation in d1. Now, we can solve this quadratic equation using the quadratic formula or any other method to find the possible values of d1.

Find the position of the third charge

After solving the quadratic equation, we will get two possible values of d1: d1₁ = 3.35 cm (approx.) d1₂ = 9.96 cm (approx.) These two positions indicate that we can place the third charge at 3.35 cm or 9.96 cm away from the positive charge (q1) along the x-axis to experience no force. Thus, the third charge should be positioned either at 3.35 cm or 9.96 cm away from the positive charge (q1) along the x-axis.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle within the realm of electrostatics. It describes the force between two stationary electric charges. The law states that the magnitude of the electrostatic force (\( F \)) between two point charges is directly proportional to the product of the magnitudes of the charges (\( q_1 \) and \( q_2 \) and inversely proportional to the square of the distance (\( r \) that separates them. This relationship can be represented by the equation: \[ F = k \frac{q_1 q_2}{r^2} \[ where \( k \) is the Coulomb's constant (~8.988×10^9 N m^2 C^-2 in vacuum). This law is crucial when solving problems involving the electric force between charges, such as determining where to place a third charge to ensure it experiences no net force from two other fixed charges.
Understanding the nuances of this relationship helps students not only solve the problems but also grasp why charges behave in a certain manner when placed in an electric field created by other charges.

Electric Force

When dealing with problems involving multiple charges, like the textbook exercise, it's essential to consider the electric force exerted on the charges. This force can be attractive or repulsive depending on the nature of the charges involved. For charges with the same sign, the electric force is repulsive; they push away from each other. For charges with opposite signs, the force is attractive; they pull towards each other.
In the particular exercise, we deal with two fixed charges of opposite signs and are tasked with finding a point where a third charge would experience no net electric force. Here, the concept of superposition is helpful; it says that the total force on a charge is the vector sum of all individual forces acting on it.
Thus, by applying Coulomb's law to each pair of charges and setting the net force on the third charge to zero, we create an equation that must be solved to find the proper placement of the third charge.

Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable x with a≠0, and it has the general form \[ ax^2 + bx + c = 0 \[ where \( a \), \( b \), and \( c \) are constants. The solutions to this equation, also known as the roots, can be found using various methods, including factoring, completing the square, graphing, or applying the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \[ In the context of our exercise, the unknown distance at which the third charge experiences no net electric force leads us to form a quadratic equation. Solving such an equation yields the potential locations for the third charge.
Remember, it's common to have two solutions, corresponding to two possible positions along the x-axis in our exercise. Each solution provides insight into the symmetrical nature of electric fields and forces around point charges, a key takeaway when you're visualizing how charges interact in space.