Problem 9 - University Physics with Modern Physics

Question

Two small plastic spheres are given positive electrical charges. When they are 15.0 \(\mathrm{cm}\) apart, the repulsive force between them has magnitude 0.220 \(\mathrm{N} .\) What is the charge on each sphere (a) if the two charges are equal and (b) if one sphere has four times the charge of the other?

Step-by-Step Solution

Answer

(a) The charge on each sphere is approximately 1.94 x 10^-8 C.

Step 1: Understanding Coulomb's Law

Coulomb's Law describes the force between two charges. The formula is \[ F = k \frac{q_1 q_2}{r^2} \] where \( F \) is the force, \( k \) is the Coulomb's constant \( 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the separation distance, converted into meters.

Step 2: Solve Part (a) - Charges are Equal

For part (a), let the charges be equal \( q_1 = q_2 = q \). Then, \[ F = k \frac{q^2}{r^2} \] Substitute \( F = 0.220 \, \mathrm{N} \) and \( r = 0.15 \, \mathrm{m} \), which gives \[ 0.220 = (8.99 \times 10^9) \frac{q^2}{0.15^2} \] Solving for \( q \), we get \[ q = \sqrt{\frac{0.220 \times 0.15^2}{8.99 \times 10^9}} \approx 1.94 \times 10^{-8} \, \mathrm{C} \].

Key Concepts

Electrical Charges

Electrical charges are fundamental properties of matter, existing in two types: positive and negative. The interaction between these charges is what gives rise to electromagnetic forces. Imagine charges like tiny invisible hands that can push or pull on objects, depending on their nature.

Similar charges, like two positive or two negative charges, repel each other. It's as if they both want to occupy the same space but can't, thus pushing away from each other. Conversely, opposite charges, meaning one positive and one negative, attract each other, much like magnets with opposite poles. This intricate dance of attraction and repulsion forms the basis for many electrical phenomena we observe in our everyday lives.

• Positive charges result from a deficiency of electrons, making other protons noticeable.
• Negative charges arise when there are excess electrons.
• The unit of charge is called a coulomb (C).

Understanding these basic interactions helps explain how two similarly charged plastic spheres in the exercise exhibit a repulsive force, pushing away from each other.

Repulsive Force

A repulsive force is a form of electrostatic force that acts to push two charged objects apart from one another. In the context of the exercise, when the two spheres have positive charges, they naturally repel each other. You can think of this force as a kind of invisible barrier preventing the spheres from getting closer.

This repulsion can be calculated using Coulomb's Law, which provides a quantitative measure of the force between the two charges. It's a key concept for students to understand, as it governs not just this problem, but many other interactions in physics and chemistry.

• Repulsive forces between like charges increase as the distance between them decreases.
• The magnitude of the force depends on both the amount of charge and the separation distance.
• As distance increases, the repulsive force decreases exponentially, meaning small changes in distance can lead to large changes in force.

This principle is fundamental in understanding how objects charged with the same type create spaces between themselves.

Coulomb's Constant

Coulomb's constant (\( k \)) is a vital number in the realm of electrostatics. It appears in Coulomb's Law and helps quantify the strength of the electrostatic force between two charges. This constant, with a value of approximately \( 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \), reflects the electrical permittivity of free space, essentially describing how electric field lines behave in a vacuum.

In simpler terms, it helps calculate how strongly two charged objects will interact with each other. It's kind of like the "weight" of an interaction, ensuring we have a consistent unit to measure these forces.

• Coulomb's constant makes it possible to predict and measure the force accurately across different distances and charges.
• This constant is derived from experimental data and is one of the fundamental constants in physics.
• It bridges the gap between theoretical physics and practical, real-world measurements, making theoretical values applicable and useful.

With this constant, we can compare and understand forces not just in one set of conditions but across any scenario involving electrical charges, grounding our understanding of the electrical nature of forces.