Question

Two carbon discs, \(1.0 \mathrm{~g}\) each, are \(1.0 \mathrm{~cm}\) apart have equal and opposite charges. If force of attraction between them is \(10^{-5} \mathrm{~N}\), the ratio of excess electrons to the total atoms on the negatively charged disc is \(\left(N_{\mathrm{A}}=6 \times 10^{23}\right)\) (a) \(2.4 \times 10^{-12}: 1\) (b) \(10^{-12}: 2.4\) (c) \(10^{12}: 2.4\) (d) \(2.4: 10^{12}\)

Short answer

The ratio of excess electrons to the total atoms on the negatively charged disc is \(2.4 \times 10^{-12}: 1\).

Step-by-step solution

Apply Coulomb's Law

Use Coulomb's law to determine the charge magnitude on the discs. Coulomb's law is given by the equation: \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \), where \(F\) is the force, \(k\) is Coulomb's constant \((8.988 \times 10^9 \, \text{N m}^2/\text{C}^2)\), \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between them.

Calculate Charge

Solve the equation for the charge magnitude \(|q|\) using the distance \(r\) and the force \(F\). Since both charges are equal and opposite, \(q_1 = q_2 = q\).

Find Number of Excess Electrons

The charge of one electron is \(e = 1.602 \times 10^{-19} \, \text{C}\). To find the number of excess electrons on the negatively charged disc, divide the total charge \(q\) computed from Step 2 by the charge of a single electron \(e\).

Determine Total Number of Atoms

The mass of one carbon atom is approximately \(12 \, \text{u}\) where \(1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg}\). Multiply this mass by Avogadro's number \(N_{\text{A}}\) to get the mass of one mole of carbon, and then find the ratio of the mass of the disc to the mass of one mole of carbon to calculate the total number of atoms in the disk.

Calculate the Ratio

Take the number of excess electrons from Step 3 and divide by the total number of atoms from Step 4 to find the required ratio.

Key concepts

Electric Charges

Electric charges are the fundamental property of particles that determine their electromagnetic interactions. In physics, electric charge is quantized, meaning it comes in discrete amounts, represented by the symbol 'e' for the elementary charge. There are two types of electric charges: positive and negative. Like charges repel each other, while opposite charges attract.

Understanding electric charge is crucial when dealing with Coulomb's Law problems, as the law describes the force between two point charges. In our exercise, carbon discs carry equal and opposite charges, which implies that one disc has an excess of electrons (negative charge), while the other lacks electrons (positive charge). The attraction between these charges is what we observe as the force of attraction between the discs.

When solving problems involving electric charges, we assume that the charges are evenly distributed over the objects. The actual distribution depends on the shape of the object and the surrounding environment. However, for simplicity in Coulomb's Law calculations, point charges or symmetric objects like the discs in our problem, allow us to treat the charge as if centered at a point.

Force of Attraction

The force of attraction between two charged objects is a fundamental concept that stems directly from Coulomb's Law. This law specifies that the magnitude of the electrostatic force of attraction or repulsion between two point-like charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.

The electrostatic force is a conservative force; this means that the work done in moving a charge is independent of the path taken, depending only on the start and end points. This property is essential in understanding electric fields and potentials. In our exercise, the force of attraction is given, and its value helps us determine the magnitude of charge on each disc.

It's important to note that Coulomb's Law applies to stationary charges. When charges are moving, additional factors such as magnetic forces come into play, which are described by other laws, such as the Lorentz force law.

Excess Electrons

Excess electrons refer to the additional electrons that an object has beyond the number that would balance out the protons' positive charge in its atoms. These extra negative charges account for the object's overall negative electric charge. In contrast, a deficiency of electrons would result in an overall positive charge.

In the context of our problem, the negatively charged carbon disc has excess electrons that create an imbalance, leading to the attraction force with the positively charged disc. Calculating the number of excess electrons involves using the charge of the disc, which we find using Coulomb's Law, and dividing this by the fundamental charge of a single electron (approximately \(1.602 \times 10^{-19} \text{C}\)).

By understanding the concept of excess electrons, we can comprehend phenomena such as static electricity and the behavior of semiconductors. This concept is a cornerstone of modern electronics, where controlling the number of electrons in a material is key to the function of devices such as transistors and diodes.