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Chapter 17: Problem 37

\(\bullet\) Electric fields in the atom. (a) Within the nucleus. What strength of electric field does a proton produce at the distance of another proton, about \(5.0 \times 10^{-15} \mathrm{m}\) away? (b) At the elec- trons. What strength of electric field does this proton produce at the distance of the electrons, approximately \(5.0 \times 10^{-10} \mathrm{m}\) away?

Short Answer

Expert verified
(a) \(5.75 \times 10^{21} \text{ N/C}\), (b) \(5.75 \times 10^{11} \text{ N/C}\).

Step by step solution

01

Understanding the Electric Field Formula

The electric field, \(E\), created by a point charge \(q\) at a distance \(r\) is given by the equation: \[ E = \frac{kq}{r^2} \] where \(k\) is the Coulomb's constant, approximately equal to \(8.99 \times 10^9 \text{ N m}^2/\text{C}^2\). In this problem, \(q\) is the charge of a proton, \(1.6 \times 10^{-19} \text{ C}\).
02

Calculating the Electric Field at the Nucleus

Substitute the known values into the electric field formula for a distance of \(5.0 \times 10^{-15} \text{ m}\):\[ E = \frac{8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \times 1.6 \times 10^{-19} \text{ C}}{(5.0 \times 10^{-15} \text{ m})^2} \] Calculate this value to find the electric field strength.
03

Compute the Result for the Nucleus

After performing the calculation, the electric field strength is found to be:\[ E \approx 5.75 \times 10^{21} \text{ N/C} \] This is the electric field at the distance of another proton in the nucleus.
04

Calculating the Electric Field at the Electrons

Now, substitute the values into the electric field formula for a distance of \(5.0 \times 10^{-10} \text{ m}\):\[ E = \frac{8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \times 1.6 \times 10^{-19} \text{ C}}{(5.0 \times 10^{-10} \text{ m})^2} \] Calculate this value to find the electric field strength.
05

Compute the Result for the Electrons

After calculating, the electric field strength is found to be:\[ E \approx 5.75 \times 10^{11} \text{ N/C} \] This is the electric field at the distance of an electron.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's constant
Coulomb's constant is a key value in electromagnetism named after the French physicist Charles-Augustin de Coulomb. It has a value of approximately \(8.99 \times 10^9 \text{ N m}^2/\text{C}^2\). This constant is fundamental when calculating the force between two charges or the electric field produced by a charge at a point in space.
  • Relates electrical force and distance
  • Used in the formula: \( F = \frac{k q_1 q_2}{r^2} \)
  • Involved in the electric field equation
Coulomb's constant helps us understand how charges interact in a vacuum.
It quantifies the strength of the electric force compared to other forces like gravity.
proton charge
The charge of a proton, a fundamental particle within an atomic nucleus, is an essential aspect of understanding electric fields. Protons, along with neutrons and electrons, are one of the building blocks of matter. The charge of a proton is denoted as \(1.6 \times 10^{-19} \text{ C}\), which is a positive charge.
  • Same magnitude but opposite sign to the charge of an electron
  • Influences the behavior of atoms and molecular structures
  • Central to chemical reactions and bonding
Understanding proton charge allows us to comprehend many aspects of electricity and magnetism in physics.
It is by the interaction of these charges that compounds and substances gain unique properties.
electric field formula
The electric field formula is a mathematical expression used to calculate the strength of the electric field (\(E\)) produced by a point charge. The formula is: \[ E = \frac{kq}{r^2} \] Where:
  • \(E\) is the electric field strength, measured in newtons per coulomb (N/C)
  • \(k\) is Coulomb's constant, \(8.99 \times 10^9 \text{ N m}^2/\text{C}^2\)
  • \(q\) is the charge, such as the proton charge \(1.6 \times 10^{-19} \text{ C}\)
  • \(r\) is the distance from the charge, in meters
This formula is essential in homework exercises and practice problems for students learning about electric fields. It helps predict how a charged particle will influence its surroundings.
Also, it's crucial for designing electronic and electrical systems, ensuring efficiency and safety in technological applications.

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Most popular questions from this chapter

\(\bullet$$\bullet\) A charge of \(-3.00 \mathrm{nC}\) is placed at the origin of an \(x y-\)coordi- nate system, and a charge of 2.00 \(\mathrm{nC}\) is placed on the \(y\) axis at \(y=4.00 \mathrm{cm} .\) (a) If a third charge, of \(5.00 \mathrm{nC},\) is now placed at the point \(x=3.00 \mathrm{cm}, y=4.00 \mathrm{cm},\) find the \(x\) and \(y\) com- ponents of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

\(\bullet\) (a) What must the charge (sign and magnitude) of a 1.45 \(\mathrm{g}\) particle be for it to remain balanced against gravity when placed in a downward-directed electric field of magnitude 650 \(\mathrm{N} / \mathrm{C}^{?}\) (b) What is the magnitude of an electric field in which the electric force it exerts on a proton is equal in magni- tude to the proton's weight?

\(\bullet\) Electrical storms. During an electrical storm, clouds can build up very large amounts of charge, and this charge can induce charges on the earth's surface. Sketch the distribution of charges at the earth's surface in the vicinity of a cloud if the cloud is positively charged and the earth behaves like a conductor.

\(\bullet\) Two iron spheres contain excess charge, one positive and the other negative. (a) Show how the charges are arranged on these spheres if they are very far from each other. (b) If the spheres are now brought close to each other, but do not touch, on the copper ball. sketch how the charges will be distributed on their surfaces. (c) In part (b), show how the charges would be distributed if both spheres were negative.

\(\bullet$$\bullet\) A total charge of magnitude \(Q\) is distributed uniformly within a thick spherical shell of inner radius \(a\) and outer radius b. (a) Use Gauss's law to find the electric field within the cavity \((r \leq a)\) . (b) Use Gauss's law to prove that the electric field outside the shell \((r \geq b)\) is exactly the same as if all the charge were concentrated as a point charge \(Q\) at the center of the sphere. (c) Explain why the result in part (a) for a thick shell is the same as that found in Example 17.10 for a thin shell. A thick shell can be viewed as infinitely many thin shells.)
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