Question

The force between a charge of \(25 \mu C\) and a charge of \(-10 \mu C\) is \(8.0 \mathrm{~N}\). What is the separation between the two charges? a) \(0.28 \mathrm{~m}\) c) \(0.45 \mathrm{~m}\) b) \(0.53 \mathrm{~m}\) d) \(0.15 \mathrm{~m}\)

Short answer

(a) 0.5 m (b) 0.53 m (c) 0.56 m (d) 0.6 m Answer: (b) 0.53 m

Step-by-step solution

Write down the given information

We are given the force between the charges \((F = 8.0 \mathrm{~N})\), the charges \((q_1 = 25 \times 10^{-6} C \textrm{ and } q_2 = -10 \times 10^{-6} C)\), and the distances \((a, b, c, d)\). We can now use the information to solve the problem using Coulomb's law.

Use Coulomb's law to find the distance between the charges

We can rearrange the formula for Coulomb's law to solve for the distance between the charges: \(r = \sqrt{\frac{k \cdot q_1 \cdot q_2}{F}}\). Now, we can plug in the given values into the formula: \(r = \sqrt{\frac{8.9875 \times 10^{9} \cdot 25 \times 10^{-6} \cdot (-10 \times 10^{-6})}{8.0}}\)

Calculate the distance

By performing the calculations, we get: \(r = \sqrt{\frac{-2.246875 \times 10^{4}}{8}}\) \(r = \sqrt{-2810.859375}\) Since force F is in N, which is a positive value and charge q_1*q_2 is negative the equation is not correct in terms of units. The correct formula should be: \(r = \sqrt{\frac{k \cdot |q_1 \cdot q_2|}{F}}\) \(r = \sqrt{\frac{8.9875 \times 10^{9} \cdot |25 \times 10^{-6} \cdot (-10 \times 10^{-6})|}{8.0}}\) \(r = \sqrt{0.28125}\) \(r ≈ 0.53\mathrm{~m}\) Thus, the separation between the two charges is approximately \(0.53\mathrm{~m}\), which corresponds to option (b).

Key concepts

Electric Charges

Electric charges are the foundation of electrostatics. They come in two types: positive and negative. Like charges repel each other while opposite charges attract. This basic property of attraction and repulsion between charges explains a multitude of phenomena, from the behavior of atoms to electrical technology.

Objects can be charged in several ways, including by friction, which transfers charge from one object to another, or by induction, where a charged object brought near a neutral one causes a separation of charges within the neutral object. A key characteristic of charge, denoted by the symbol 'q', is that it is conserved. In any system, the total charge remains constant, and this principle applies to the interactions between charges as well.

Electrostatic Force

Electrostatic force is the force that electric charges exert on each other. This force can act over a distance, allowing charged objects to interact without directly touching. It's one of the fundamental forces in physics and is governed by Coulomb's Law.

The electrostatic force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between their centers. It is immensely powerful on a small scale and is responsible for holding atoms and molecules together. However, as the distance between charges increases, the force decreases sharply, which is why we don’t feel these forces in our everyday life.

Coulomb's Law Formula

Coulomb's Law formula quantitatively describes the electrostatic force between two charges. The law was named after Charles-Augustin de Coulomb, who introduced the formula in the 18th century. The formula is as follows:
\[F = k\frac{|q_1 \times q_2|}{r^2}\]
Here, \(F\) represents the force between charges, \(k\) is the Coulomb's constant (\(8.99 \times 10^9 \mathrm{N m^2/C^2}\)), \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the distance between the centers of the two charges. The absolute value is used because the force is always positive, indicating it has a magnitude regardless of the charges being like or opposite.

It's crucial to use the distance \(r\) accurately as it greatly affects the calculated force—doubling the distance decreases the force by a factor of four, illustrating the inverse square nature of the relationship.