The electron in a hydrogen atom is \(52.9 \mathrm{pm}\) from the proton. What's the proton's electric field strength at this distance?

Coulomb's Law is a fundamental principle in physics that describes the force between two charges. Named after Charles-Augustin de Coulomb, this law states that the magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The mathematical representation of Coulomb's Law is:
\[ F = k \frac{Q_1 \times Q_2}{r^2} \]
Here, \( F \) is the force between the charges, \( Q_1 \) and \( Q_2 \) are the amounts of the charges, \( r \) is the distance between the centers of the two charges, and \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \) Nm²/C²). This law is essential for understanding interactions at the atomic level, such as between the proton and electron in a hydrogen atom. Coulomb's Law forms the basis for many calculations involving electric forces and fields in physics.

The electric field is a vector field that surrounds electric charges and exerts force on other charges within that field. The strength of the electric field at a point in space, often designated as \( E \), is defined as the force per unit charge that a test charge would experience at that point. The formula for electric field strength derived from Coulomb’s Law is given by:
\[ E = k \frac{Q}{r^2} \]
Where \( E \) is the electric field strength, \( Q \) is the charge creating the electric field, \( r \) is the distance from the charge to the point of interest, and \( k \) is Coulomb's constant. For example, in the case of a proton and an electron in an atom, the electric field strength at any point surrounding the proton can be calculated using the above formula, if the charge of the proton and the distance from it are known. Since the charge of a proton is the elementary charge \( e = 1.602 \times 10^{-19} C \), this value is used for \( Q \) in the formula.

Unit conversion is a critical skill in science and engineering, as it ensures measurements are comparable and calculations are accurate. Working with the electric field formulas often requires converting units of distance and charge. For instance, the exercise provided involves converting picometers (pm) to meters (m), which is essential for consistency with the SI units used in Coulomb’s Law. The conversion from picometers to meters involves multiplying the picometer value by \( 10^{-12} \), since there are \( 10^{12} \) picometers in a meter. The ability to convert units accurately allows for proper substitution into the electric field formula and ensures the final result is expressed in the correct units, such as newtons per coulomb (N/C) for electric field strength. It's important to remember these conversions and to carry out the appropriate mathematical operations to make certain that the results are not only numerically correct but also make physical sense within the context of the given problem.