Problem 28 - University Physics with Modern Physics

Question

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 \(\mathrm{V}\) and \(12.0 \mathrm{V} / \mathrm{m},\) respectively. (Take the potential to be zero 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 point charge?

Step-by-Step Solution

Answer

Distance is approximately 0.415 m, charge is about \( 2.30 \times 10^{-10} \mathrm{C} \), field points away from charge.

Step 1: Understanding the Relationship between Potential and Electric Field

The electric potential \( V \) at a distance \( r \) from a point charge \( Q \) is given by \( V = \frac{kQ}{r} \), where \( k \) is Coulomb's constant (\( 8.99 \, \times \, 10^9 \, \mathrm{N} \, \mathrm{m}^2/\mathrm{C}^2 \)). The electric field \( E \) is given by \( E = \frac{kQ}{r^2} \). We have \( V = 4.98 \, \mathrm{V} \) and \( E = 12.0 \, \mathrm{V}/\mathrm{m} \).

Step 2: Relating V and E to Find Distance

Using the given equations, we have two equations: \(\frac{kQ}{r} = 4.98 \) and \( \frac{kQ}{r^2} = 12.0 \). Dividing the potential equation by the electric field equation \( \frac{kQ/r}{kQ/r^2} = \frac{4.98}{12.0} \), we can solve for \( r \) as: \( r = \frac{4.98}{12.0} \approx 0.415 \, \mathrm{m} \).

Step 3: Finding the Magnitude of the Point Charge

Using \( V = \frac{kQ}{r} \), substitute \( r = 0.415 \, \mathrm{m} \) and \( V = 4.98 \, \mathrm{V} \) to find \( Q \). Rearranging gives \( Q = \frac{Vr}{k} = \frac{4.98 \, \times \, 0.415}{8.99 \, \times \, 10^9} \approx 2.30 \, \times \, 10^{-10} \, \mathrm{C} \).

Step 4: Determining the Direction of the Electric Field

The electric field points away from positive charges and towards negative charges. Since only the magnitudes are provided and not whether the field or potential is positive or negative, with no additional charges nearby, we typically assume the point charge to be positive, hence the field points away from the charge.

Key Concepts

Point Charge

A point charge is a hypothetical charge that is considered to be concentrated at a single point in space. This notion is incredibly useful in physics, especially when studying the behavior of electric fields and potentials, as it simplifies calculations. Point charges are assumed to have no physical size, which allows physicists to analyze their effects without worrying about the complexities that come with real objects with volumes.

Point charge is a vital concept in electromagnetism because:

• It allows us to easily calculate electric fields and potentials.
• It lays the foundation for understanding more complex charge distributions.
• It is a simplified model that still delivers high precision in predicting the real-world behavior of more complex systems.

In practical terms, when dealing with a point charge, the electric potential and field can be described using simple equations. These relationships reveal how the charge influences the space around it, giving insight into how it interacts with other charges in its vicinity.

Coulomb's Law

Coulomb's Law is a fundamental principle that describes how two point charges interact in an electric field. It states that the force 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 law 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 amounts of the two charges,
• \( r \) is the distance between the charges, and
• \( k \) is Coulomb’s constant, approximately \( 8.99 \times 10^9 \mathrm{N} \mathrm{m}^2/\mathrm{C}^2 \).

Coulomb's Law is instrumental for understanding electrostatic interactions and is the electrostatic equivalent of Newton's law of universal gravitation. The law explains:

• How electric forces can both attract and repel, unlike gravitational forces which only attract.
• The importance of distance as even a small increase can greatly reduce the force, which is why charged objects need to be close to significantly interact.
• The role of the medium between the charges, as this law assumes a vacuum or air, but different materials can influence the force.

Electric Field Direction

The direction of an electric field is conceptually simple but crucial to understanding the nature of electric interactions. It is defined as the direction that a positive test charge would move if placed within the field. This means that the electric field always points away from positive charges and toward negative charges.

Understanding the direction of electric fields helps in:

• Predicting how electrons will move, as they are negatively charged and thus move opposite to the field direction.
• Designing electric circuits and components by informing the placement of positive and negative terminals.
• Understanding the behavior of various materials in electric fields, particularly conductive versus insulative materials.

Let's consider our exercise: if we assume that the point charge is positive, the electric field will point away from it. This aligns with the observations about how electric fields operate around point charges, providing straightforward predictions about interactions with other charges or test charges placed in the vicinity.