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Chapter 23: Problem 22

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are \(4.98 \mathrm{~V}\) and \(16.2 \mathrm{~V} / \mathrm{m}\) respectively. (Take \(V=0\) at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the noint charge?

Short Answer

Expert verified
The distance to the point charge is 0.364 m, the magnitude of the charge is \(2.02 脳 10^-10 C\), and the electric field is directed away from the point charge.

Step by step solution

01

Identify Relevant Formulas

Let's start by identifying the relevant formulas for this problem. The potential \(V\) and the electric field \(E\) at a distance \(r\) due to a point charge \(q\) are given respectively by: \(V = \frac{kq}{r}\) and \(E = \frac{kq}{r^2}\) Here, \(k\) is Coulomb's constant, and it is equal to \(8.99 脳 10^9 N m^2/C^2\).
02

Calculate Distance to the Point Charge

We know \(V\) as well as \(E\). We can use the ratio of \(E/V\) for calculation the distance \(r\). Using the formulas for \(V\) and \(E\) above, this ratio equals to: \(E/V = r = \frac{1}{k} 脳 16.2 V/m 梅 4.98 V = 0.364 m\) Thus, the distance to the point charge is 0.364 m.
03

Calculate Magnitude of the Charge

Now, to determine the magnitude of the charge \(q\), we substitute the calculated distance \(r\) into the formula for \(V\), solving for \(q\): \(q = \frac{Vr}{k} = 4.98 V 脳 0.364 m 梅 8.99 脳 10^9 N m^2/C^2 = 2.02 脳 10^-10 C\) So, the magnitude of the charge is \(2.02 脳 10^-10\) Coulombs.
04

Determine Direction of Electric Field

Finally, the direction of the electric field in a region of space is the direction that a positive test charge would be pushed if placed at that location. Since the potential \(V\) is positive, the initial point charge \(q\) must be positive. So the electric field from a positive charge is always directed away from the charge. Thus, the electric field here is also directed away from the point charge.

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Key Concepts

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

Coulomb's Law
Understanding Coulomb's Law is essential when studying electric forces between two point charges. It states that the magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitude of charges and inversely proportional to the square of the distance between them.

It is mathematically expressed as
\[ F = k \frac{|q_1 q_2|}{r^2} \]
Where:
  • \( F \) is the magnitude of the force between the charges,
  • \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
  • \( r \) is the distance between the centers of the two charges,
  • \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2) \).
This fundamental principle allows us to calculate the force and also implies that like charges repel, while unlike charges attract each other. The 'point charge' assumption presumes the charge occupies no space, which is a useful abstraction for calculations.
Electric Potential
Electric potential at a point in space, often referred to as voltage (\( V \)), is the amount of electric potential energy per unit charge at a specific location, and it's influenced by the presence of electric charges.

For a point charge, the electric potential is given by the formula:
\[ V = \frac{kq}{r} \]
where:
  • \( V \) is the electric potential,
  • \( k \) is Coulomb's constant,
  • \( q \) is the magnitude of the charge creating the potential,
  • \( r \) is the distance from the charge.
The potential provides insight into how much work would be done by or against an electric field in moving a unit positive charge from infinity to that point. The higher the potential, the greater the work required. Electric potential is scalar, which means it doesn't have a direction but rather a value at each point in space.
Magnitude of Charge
The magnitude of charge refers to the size of the electric charge on an object, quantified in Coulombs (\( C \)). It represents the surplus or deficit of electrons on the object, with the fundamental charge of one electron being approximately \( 1.6 \times 10^{-19} \) Coulombs.

Determining the magnitude of the charge is crucial when assessing the effects of a charge on nearby objects and fields. The charge magnitude factors into equations governing electrostatic forces, electric fields, and potentials. It's also central to laws like Coulomb's Law, as seen where charge magnitude directly impacts the strength of the force between charges.
Direction of Electric Field
The direction of an electric field is conventionally defined as the path a positive test charge would take if placed within the field. In other words, the electric field vector at a point in space points in the direction that a positive charge would feel a force.

For a positive point charge, the electric field points radially away from the charge, whereas for a negative point charge, it points radially towards the charge. The magnitude diminishes with distance but the direction remains radial. Therefore, knowing both the sign of the charge and the configuration of the field allows one to make predictions about the behavior of other charges within that field.

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Most popular questions from this chapter

A metal sphere with radius \(r_{a}\) is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius \(\eta_{b}\). There is charge \(+q\) on the inner sphere and charge \(-q\) on the outer spherical shell. (a) Calculate the potential \(V(r)\) for (i) \(rn\). (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take \(V\) to be zero when \(r\) is infinite. (b) Show that the potential of the inner sphere with respect to the outer is $$ V_{a b}=\frac{q}{4 \pi \epsilon_{0}}\left(\frac{1}{r_{a}}-\frac{1}{n_{b}}\right) $$ (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude \(E(r)=\frac{V_{a b}}{\left(1 / r_{a}-1 / r_{b}\right)} \frac{1}{r^{2}}\) (d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance \(r\) from the center, where \(r>r_{b} .\) (e) Suppose the charge on the outer sphere is not \(-q\) but a negative charge of different magnitude, say \(-Q .\) Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different

BIO Electrical Sensitivity of Sharks. Certain sharks can detect an electric field as weak as \(1.0 \mu \mathrm{V} / \mathrm{m}\). To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5 V AA battery across these plates, how far apart would the plates have to be?

(a) How much excess charge must be placed on a copper sphere \(25.0 \mathrm{~cm}\) in diameter so that the potential of its center is \(3.75 \mathrm{kV} ?\) Take the point where \(V=0\) to be infinitely far from the sphere, (b) What is the potential of the sphere's surface?

A point charge \(+8.00 \mathrm{nC}\) is on the \(-x\) -axis at \(x=-0.200 \mathrm{~m}\) and a point charge \(-4.00 \mathrm{nC}\) is on the \(+x\) -axis at \(x=0.200 \mathrm{~m}\). (a) In addition to \(x=\pm \infty\), at what point on the \(x\) -axis is the resultant field of the two charges equal to zero? (b) Let \(V=0\) at \(x=\pm \infty\), At what two other points on the \(x\) -axis is the total electric potential due to the two charges equal to zero? (c) Is \(E=0\) at either of the points in part (b) where \(V=0 ?\) Explain.

A particle with charge \(+7.60 \mathrm{nC}\) is in a uniform electric field directed to the left. Another force, in addition to the electric force, acts on the particle so that when it is released from rest, it moves to the right. After it has moved \(8.00 \mathrm{~cm}\), the additional force has done \(6.50 \times 10^{-5} \mathrm{~J}\) of work and the particle has \(4.35 \times 10^{-5} \mathrm{~J}\) of kinetic energy. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of the electric field?
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