Question

Use Coulomb's law, \(F=k Q_{1} Q_{2} / d^{2},\) to calculate the electric force on an electron \(\left(Q=-1.6 \times 10^{-19} \mathrm{C}\right)\) exerted by a single proton if the particles are \(0.53 \times 10^{-10} \mathrm{~m}\) apart. The constant \(k\) in Coulomb's law is \(9.0 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\). (The unit abbreviated \(\mathrm{N}\) is the newton, the SI unit of force.)

Short answer

The electric force exerted by a single proton on an electron, when the particles are \(0.53 \times 10^{-10} m\) apart, is approximately \(5.12 \times 10^{-9} N\).

Step-by-step solution

Identify given information

We are given the following information: - Charge of an electron: \(Q_1 = -1.6 \times 10^{-19} C\) - Charge of a proton: \(Q_2 = 1.6 \times 10^{-19} C\) (we know the magnitude of the charge is equal to the charge of an electron, but with the opposite sign) - Distance between the electron and proton: \(d = 0.53 \times 10^{-10} m \) - Coulomb's constant: \(k = 9.0 \times 10^9 N \cdot m^2 / C^2\)

Apply Coulomb's Law formula

Using the given charges, distance, and Coulomb's constant, we can now find the electric force between the electron and the proton by applying Coulomb's Law: \[ F = \frac{k \cdot Q_1 \cdot Q_2}{d^2} \]

Substitute the given values

Now substitute the values in the formula: \[ F = \frac{(9.0 \times 10^9 N \cdot m^2 / C^2)(-1.6 \times 10^{-19} C)(1.6 \times 10^{-19} C)}{(0.53 \times 10^{-10} m)^2} \]

Solve for the electric force

Simplify the expression and solve for the electric force: \[ F = \frac{(9.0 \times 10^9)(16 \times 10^{-38})}{2.809 \times 10^{-20}} \] \[ F = \frac{14.4 \times 10^{-28}}{2.809 \times 10^{-20}} \] \[ F = 5.12 \times 10^{-9} N \] The electric force exerted by a single proton on an electron, when the particles are \(0.53 \times 10^{-10} m\) apart, is approximately \(5.12 \times 10^{-9} N\).

Key concepts

electric force calculation

Electric force calculation is crucial for understanding interactions between charged particles. The calculation of electric force utilizes Coulomb's Law, which describes the electrostatic force exerted between two static electric charges. According to this law, the magnitude of the electric force, \( F \), between two point charges \( Q_1 \) and \( Q_2 \), separated by a distance \( d \), is calculated as:
\[\ F = \frac{k \cdot Q_1 \cdot Q_2}{d^2}\]
where:

• \( k \) is the Coulomb constant \( 9.0 \times 10^9 \text{ N} \cdot \text{m}^2 / \text{C}^2 \).
• \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges.
• \( d \) is the distance between the charges.

The formula signifies that the electric force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. To calculate the force, simply substitute the known values into the equation and solve. With this understanding, calculating electric force becomes a straightforward process that allows the precise determination of how charged particles will interact.

electron and proton interaction

Electrons and protons are fundamental components of an atom, with each possessing distinct electric charges. Electrons, denoted by the charge \(-1.6 \times 10^{-19}~\text{C}\), are negatively charged, while protons carry an equal but opposite positive charge \(+1.6 \times 10^{-19}~\text{C}\). This charge balance is essential in atomic structures, defining much of the behavior of atoms.
Interactions between electrons and protons are governed by the principles of electromagnetism. These include:

• A negative and positive charge will attract each other, which is clearly observed in the proton-electron interaction.
• This attraction results in a binding force within an atom, maintaining the stability of the atomic structure.

When calculating the force of attraction between an electron and a proton using Coulomb's Law, it's crucial to consider their distance apart. As demonstrated in the example, a significant electric force of attraction exists even at a small scale, illustrating the powerful nature of electrostatic interactions.

physics problem solving

Solving physics problems often requires a structured approach to break down and solve complex numerical scenarios. Here, it's important to:
1. **Identify and organize given information:** Start by listing all known values such as charge magnitude, distance, and any constants involved. For instance, know the electron charge \(-1.6 \times 10^{-19}\) and proton charge \(+1.6 \times 10^{-19}\).
2. **Consider relevant formulas:** Use appropriate physical laws like Coulomb's Law to derive relationships fundamental to the problem. This involves setting up the formula correctly.
3. **Substitute and compute:** Carefully substitute known values into the formula. Perform calculations step-by-step to avoid errors. Taking our example, substituting the given charges and distance into the formula:
\[\ F = \frac{(9.0 \times 10^9)(-1.6 \times 10^{-19})(1.6 \times 10^{-19})}{(0.53 \times 10^{-10})^2}\]
4. **Evaluate results:** After computing, verify the result makes sense physically. In our problem, the calculated force \(5.12 \times 10^{-9}~\text{N}\) reflects expected forces between atomic particles.
Approaching problems with structured methods ensures accurate and understandable solutions, fostering better learning and application of physics concepts.