Question

\( \mathrm{~A}-4.0-\mu \mathrm{C}\) charge lies \(20.0 \mathrm{~cm}\) to the right of a \(2.0-\mu \mathrm{C}\) charge on the \(x\) -axis. What is the force on the \(2.0-\mu C\) charge?

Short answer

Question: Determine the force exerted on the 2.0-μC charge, given two charges of -4.0-μC and 2.0-μC are separated by a distance of 20.0 cm. Answer: The force exerted on the 2.0-μC charge is -1.798 x 10^3 N. The negative sign indicates that the force is attractive, meaning the charges are opposite in nature and are pulling each other together.

Step-by-step solution

Convert distances to meters

Given the distance is \(20.0\) cm, we will convert it to meters by dividing by 100: \(20.0 \mathrm{~cm} = \frac{20.0}{100} \mathrm{~m} = 0.200 \mathrm{~m}\)

Identify charges and distance

We have two charges, \(q_1 = -4.0-\mu C\) and \(q_2 = 2.0-\mu C\), and the distance between them is \(r = 0.200 \mathrm{~m}\).

Apply Coulomb's Law formula

We will use the following formula: \(F = k\frac{q_1q_2}{r^2}\). Plugging in the charges, distance, and constant values: \(F = (8.99 \times 10^9 \ Nm^2C^{-2})\frac{(-4.0 \times 10^{-6} \ C)(2.0 \times 10^{-6} \ C)}{(0.200 \ m)^2}\)

Calculate the force

Now, we will perform the calculations to find the force: $F = (8.99 \times 10^9 \ Nm^2C^{-2})(-8.0 \times 10^{-12} \ C^2) \div (0.0400 \ m^2) \\ F = (-71.92 \times 10^{-3} \ Nm^2) \div (0.0400 \ m^2) \\ F = -1.798 \times 10^3 \ N$ The force on the \(2.0-\mu C\) charge is \(-1.798 \times 10^3 \ N\). The negative sign indicates that the force is attractive, meaning the charges are opposite in nature and are pulling each other together.

Key concepts

Electric Charge

Understanding electric charges is fundamental to studying electric forces and how they interact. An electric charge is a property of matter that allows it to experience a force when near other electrically charged matter. There are two types of electric charges: positive (\texttt{+}) and negative (\texttt{-}). Like charges repel each other while opposite charges attract. The unit of electric charge is the coulomb (C). In our problem, we dealt with microcoulombs (\texttt{\(\mu\)C}), which are one millionth of a coulomb (\(10^{-6}\) C).

It is crucial to know that electric charges can be measured by static (stationary) or dynamic (flowing, as in current) means, and they can exist on objects as excess or deficit of electrons. The balance or imbalance of these charges leads to the presence of an electric force.

Electric Force

The electric force is the attraction or repulsion between electric charges. It is described quantitatively by Coulomb's Law. This law states that the magnitude of the electric force (\texttt{F}) between two point charges is directly proportional to the product of the magnitudes of charges (\texttt{\(q_1\)} and \texttt{\(q_2\)}) and inversely proportional to the square of the distance (\texttt{\(r\)}) between them. The constant of proportionality is denoted by \texttt{k}, known as Coulomb's constant (\(8.99 \times 10^9 \texttt{ Nm}^2\texttt{C}^{-2}\)).

Mathematically, Coulomb's Law is expressed as \texttt{\(F = k \frac{q_1q_2}{r^2}\)}. In the problem presented, we used this formula to calculate the electric force between two charges. It's important to note that the direction of the force is not given by this equation and depends on the nature of the charges: if the product of charges (sign-wise) is positive, the forces are repulsive; if negative, they are attractive, as was the case in our problem.

Physics Problem Solving

Solving physics problems requires a systematic approach that often involves several key steps: understanding the problem, visualizing it, identifying the relevant concepts, and applying mathematical equations.

In our example, we followed these steps to solve for the electric force between two charges. First, we converted units to ensure consistency. Then, the problem was assessed, identifying the two charges and the distance between them. Next, we applied the appropriate physics principle, Coulomb's Law. Finally, we executed the calculations to arrive at a quantifiable force.

When solving physics problems, it's vital to keep track of units, signs (which indicate directions), and constants. Moreover, one should check the final answer to see if it makes physical sense. For instance, in our problem, the negative force indicated an attractive interaction between the charges, consistent with their opposite natures. This logical check can be an invaluable tool for students to confirm their understanding of the concepts at play.