Problem 25 - College Physics

Question

At what distance from a proton is the magnitude of its electric field \(1.0 \times 10^{5} \mathrm{~N} / \mathrm{C} ?\)

Step-by-Step Solution

Answer

The distance is approximately \(3.79 \times 10^{-7}\) meters.

Step 1: Understand the problem

We're asked to find the distance from a proton at which the magnitude of its electric field is known. We can use the formula for the electric field produced by a point charge to solve this problem.

Step 2: Electric field formula

The formula for the electric field \(E\) due to a point charge \(q\) at a distance \(r\) is given by:\[ E = \frac{k \cdot q}{r^2} \]Where \(k\) is Coulomb's constant \((8.99 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2)\) and \(q\) is the charge of a proton \((1.6 \times 10^{-19} \text{ C})\).

Step 3: Rearrange the formula

To find the distance \(r\), we need to rearrange the formula:\[ r^2 = \frac{k \cdot q}{E} \]\[ r = \sqrt{\frac{k \cdot q}{E}} \]

Step 4: Substitute the known values

Now substitute the known values into the equation:\[ r = \sqrt{\frac{8.99 \times 10^9 \cdot 1.6 \times 10^{-19}}{1.0 \times 10^5}} \]

Step 5: Perform the calculation

Calculate the value using the substituted numbers:\[ r = \sqrt{\frac{1.4384 \times 10^{-9}}{1.0 \times 10^5}} \]\[ r = \sqrt{1.4384 \times 10^{-14}} \]\[ r \approx 3.79 \times 10^{-7} \text{ m} \]

Key Concepts

Proton

A proton is a subatomic particle found in the nucleus of an atom. It carries a positive electrical charge, equal in magnitude but opposite in sign to that of an electron. This positive charge is crucial in atomic structure and electrical interactions.
The presence of protons in the nucleus contributes to most of an atom's mass, along with neutrons, which are neutrally charged particles. Protons play a vital role in determining an element's identity, as the number of protons in an atom's nucleus defines the atomic number and, consequently, the element itself. For instance, hydrogen, the lightest and most abundant element, has one proton.
Protons have a significant influence on electrostatic interactions. Because they are positively charged, they exert electric forces on other charges, an effect we calculate using the electric field produced by this charge.

Coulomb's constant

Coulomb's constant, denoted as \(k\), is a key factor in electrostatic calculations. It is a proportionality constant in Coulomb's law, which describes the force between two point charges separated by a distance.
The value of Coulomb's constant is \(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\). This constant allows us to calculate the electric force between charges in free space and is essential in the formula for determining electric fields.
The constant illustrates the strength of the electric force; the larger the constant, the stronger the force between charges for a given distance and charge magnitude.

Electric field formula

The electric field formula is used to describe the electric field \(E\) caused by a point charge \(q\) at a specific distance \(r\) from the charge. The formula is given by:

• \( E = \frac{k \cdot q}{r^2} \) - This equation calculates the electric field produced by a point charge.
• \(k\) is Coulomb’s constant, and \(q\) is the charge of the proton.

The electric field describes how a charge distributes its influence in the space surrounding it.
It can be thought of as the force experienced by a positive test charge placed in the vicinity of the source charge (here, the proton). Understanding the electric field notion is crucial in analyzing how charges interact in different environments and is fundamental in many areas of physics and engineering.

Distance calculation

To calculate the distance \(r\) from a proton where the electric field is a given magnitude, you need to use the rearranged electric field formula. Here’s the step-by-step way to do it:

• Start with the formula of electric field: \( E = \frac{k \cdot q}{r^2} \).
• Rearrange it to solve for \(r\): \( r^2 = \frac{k \cdot q}{E} \).
• Take the square root: \( r = \sqrt{\frac{k \cdot q}{E}} \).

Substitute the known values into the formula:

• Set \(k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\), \(q = 1.6 \times 10^{-19} \, \text{C}\), and \(E = 1.0 \times 10^5 \, \text{N/C}\).
• Calculate it: \( r = \sqrt{\frac{8.99 \times 10^9 \cdot 1.6 \times 10^{-19}}{1.0 \times 10^5}} \).
• The result will give you the distance \(r\), approximately \(3.79 \times 10^{-7} \, \text{m}\).

This calculation shows how electric force diminishes with distance, highlighting the principle that electric fields become weaker as you move away from the charge.