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Chapter 21: Problem 10

Two point charges are placed on the \(x\) -axis as follows: Charge \(q_{1}=+4.00 \mathrm{nC}\) is located at \(x=0.200 \mathrm{~m},\) and charge \(q_{2}=+5.00 \mathrm{nC}\) is at \(x=-0.300 \mathrm{~m} .\) What are the magnitude and direction of the total force exerted by these two charges on a negative point charge \(q_{3}=-6.00 \mathrm{nC}\) that is placed at the origin?

Short Answer

Expert verified
The magnitude of the total force exerted by these two charges on the negative point charge \(q_3\) is given by \(F_{net} = |F_2 - F_1|\). The direction is towards the side of the charge that exerts the larger force.

Step by step solution

01

Identify knowns and unknowns

Let's identify what we know and what we need to find out. We know the position and charge for all three charges. The unknowns are the magnitude and direction of the total force exerted by \(q_1\) and \(q_2\) on \(q_3\).
02

Determine forces individually

Let's use Coulomb's law to determine the forces exerted by \(q_1\) and \(q_2\) on \(q_3\) individually. Coulomb's law formula is \(F = K \cdot \frac{|q_1 \cdot q_2|}{r^2}\). Another thing to consider is the force direction. \(q_1\) and \(q_3\) are both positive charges, so they repel each other, and the force is to the right. \(q_2\) and \(q_3\) are both negative charges, so they also repel each other, and the force is to the left.
03

Calculate individual forces

Now plug in the known values into the Coulomb's law. Here, \(K\) is Coulomb's constant (\(K = 9.00 \times 10^9 \, N \cdot m^2/C^2\)), \(q_i\) is the charge, and \(r\) is the distance between the charges. So we calculate, \(F_1 = K \cdot \frac{|q_1 \cdot q_3|}{r_1^2}\) and \(F_2 = K \cdot \frac{|q_2 \cdot q_3|}{r_2^2}\).
04

Calculate total force

Now, to find the total force, since \(F_1\) and \(F_2\) have opposite directions, we subtract them to get \(F_{net} = |F_2 - F_1|\). Also, the direction of the net force will be towards the charge inducing larger force.
05

Find direction of total force

Observe the magnitude of resultant force and decide the direction of force. If \(|F_2| > |F_1|\), the force is towards the left. Otherwise, it is towards the right.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Charge
Electric charge is one of the fundamental properties of matter, intimately related to electromagnetic interactions. It's classified into two types - positive and negative. Like charges repel each other, while unlike charges attract. The unit of electric charge is the coulomb (C). Subatomic particles carry charges, with electrons having a negative charge and protons a positive charge. The conservation of charge principle states that the total charge in an isolated system remains constant no matter what changes take place within the system.

Charges exert forces on each other, which bring about the vast array of observable phenomena in electricity and magnetism. In our exercise example, we see point charges, which can be thought of as idealized charges concentrated at a single point in space. This concept simplifies the problem by reducing the complex distribution of charges to manageable calculations using Coulomb's law.
Electric Force
Electric force is the push or pull that charged objects exert on each other. The magnitude of this force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, as stated by Coulomb's law. Mathematically, it's expressed as \(F = K \cdot \frac{|q_1 \cdot q_2|}{r^2}\), where \(F\) is the force, \(K\) is Coulomb's constant (\(9.00 \times 10^9 \, N \cdot m^2/C^2\)), \(|q_1 \cdot q_2|\) is the absolute product of the charges, and \(r\) is the distance between the charges.

In the context of our exercise, we're using Coulomb's law to calculate the forces between each pair of charges to understand the net force acting on a charge. By considering both magnitude and direction, we ensure accurate and complete solutions to problems involving electrostatic forces.
Electrostatics
Electrostatics is the branch of physics that studies electric charges at rest. As opposed to electrodynamics, which involves moving charges and varying fields, electrostatics focuses on static charges and constant electric fields. Central to electrostatics is the concept that force is exerted without physical contact; this action-at-a-distance is due to electric fields generated by static charges.

An electric field exerts forces on charges within its influence, giving us a useful way to visualize and calculate how charged objects behave when placed in such fields. The problem we're looking at involves electrostatic forces as all charges are stationary. By understanding electrostatic principles, one can predict how the negative point charge in our exercise will move in response to the forces exerted by the other two stationary charges.

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Most popular questions from this chapter

Two small aluminum spheres, each having mass \(0.0250 \mathrm{~kg}\), are separated by \(80.0 \mathrm{~cm}\). (a) How many electrons does each sphere contain? (The atomic mass of aluminum is \(26.982 \mathrm{~g} / \mathrm{mol}\), and its atomic number is \(13 .\) ) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude \(1.00 \times 10^{4} \mathrm{~N}\) (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the electrons in each sphere does this represent?

(a) What must the charge (sign and magnitude) of a \(1.45 \mathrm{~g}\) particle be for it to remain stationary when placed in a downwarddirected electric field of magnitude \(650 \mathrm{~N} / \mathrm{C} ?\) (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitude to its weight?

An American penny is \(97.5 \%\) zinc and \(2.5 \%\) copper and has a mass of \(2.5 \mathrm{~g}\). (a) Use the approximation that a penny is pure zinc, which has an atomic mass of \(65.38 \mathrm{~g} / \mathrm{mol},\) to estimate the number of electrons in a penny. (Each zinc atom has 30 electrons.) (b) Estimate the net charge on all of the electrons in one penny. (c) The net positive charge on all of the protons in a penny has the same magnitude as the charge on the electrons. Estimate the force on either of two objects with this net magnitude of charge if the objects are separated by \(2 \mathrm{~cm}\). (d) Estimate the number of leaves on an oak tree that is 60 feet tall. (e) Imagine a forest filled with such trees, arranged in a square lattice, each \(10 \mathrm{~m}\) distant from its neighbors. Estimate how large such a forest would need to be to include as many leaves as there are electrons in one penny. (f) How does that area compare to the surface area of the earth?

The ammonia molecule \(\left(\mathrm{NH}_{3}\right)\) has a dipole moment of \(5.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} .\) Ammonia molecules in the gas phase are placed in a uniform electric field \(\vec{E}\) with magnitude \(1.6 \times 10^{6} \mathrm{~N} / \mathrm{C}\). (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to \(\vec{E}\) from parallel to perpendicular? (b) At what absolute temperature \(T\) is the average translational kinetic energy \(\frac{3}{2} k T\) of a molecule equal to the change in potential energy calculated in part (a)? (Note: Above this temperature, thermal agitation prevents the dipoles from aligning with the electric field.)

A semicircle of radius \(a\) is in the first and second quadrants, with the center of curvature at the origin. Positive charge \(+Q\) is distributed uniformly around the left half of the semicircle, and negative charge \(-Q\) is distributed uniformly around the right half of the semicircle (Fig. P21.84). What are the magnitude and direction of the net electric field at the origin produced by this distribution of charge?
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