Problem 26 - University Physics with Modern Physics

Question

A particle has charge \(-3.00 \mathrm{nC}\) . (a) Find the magnitude and direction of the electric field due to this particle at a point 0.250 m directly above it. (b) At what distance from this particle does its electric field have a magnitude of 12.0 \(\mathrm{N} / \mathrm{C} ?\)

Step-by-Step Solution

Answer

(a) 431.04 N/C, downward; (b) 0.750 m.

Step 1: Understand the Formula for Electric Field

The magnitude of the electric field \(E\) due to a point charge \(q\) is given by the formula \(E = \frac{k |q|}{r^2}\), where \(k\) is Coulomb's constant \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\), \(q\) is the charge in coulombs, and \(r\) is the distance from the charge in meters.

Step 2: Calculate Electric Field Magnitude for Part (a)

Convert the charge from nanocoulombs to coulombs: \(-3.00 \, \text{nC} = -3.00 \times 10^{-9} \, \text{C}\).Using the formula for electric field, \(E = \frac{(8.99 \times 10^9) \times 3.00 \times 10^{-9}}{(0.250)^2}\).Calculate the magnitude as \(E \approx 431.04 \, \text{N/C}\).

Step 3: Determine the Direction for Part (a)

Since the charge is negative, the electric field direction is toward the charge. Therefore, the electric field at the point 0.250 m above the charge is directed downward.

Step 4: Set Up Equation for Part (b)

We need to find the distance \(r\) such that the electric field \(E = 12.0 \, \text{N/C}\).Use the formula \(12 = \frac{(8.99 \times 10^9) \times 3.00 \times 10^{-9}}{r^2}\).

Step 5: Solve for Distance in Part (b)

Rearrange the equation: \(r^2 = \frac{(8.99 \times 10^9) \times 3.00 \times 10^{-9}}{12}\).Solve for \(r\) to find \(r \approx 0.750 \, \text{m}\).

Key Concepts

Coulomb's Law

Coulomb's Law is fundamental in calculating the electric forces between point charges, as well as the electric field created by such charges. It states that the magnitude of the electric force \(F\) between two point charges is directly proportional to the product of the magnitudes of the charges \(q_1\) and \(q_2\), and inversely proportional to the square of the distance \(r\) between them.
The formula is expressed as \[ F = \frac{k |q_1 q_2|}{r^2}, \]
where \(k\) is Coulomb’s constant, \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\).

While Coulomb's Law calculates force, it is also crucial for finding the electric field \(E\) created by a point charge \(q\).

• To find the electric field, we modify the formula to \(E = \frac{k |q|}{r^2}\).
• This describes how a point charge influences a region in space.
• The further away the point of observation is from the charge, the weaker the field becomes, as it diminishes with the square of the distance.

Point Charge

A point charge is a charge concentrated at a single point in space. This concept is an idealized model used to simplify calculations in electrostatics.
Such charges can represent small charged bodies where their size is negligible compared to the distance from the point of interest.

The charge in the original exercise is \(-3.00 \, \text{nC}\) (nanocoulombs, where \(1 \, \text{nC} = 10^{-9} \, \text{C}\)). When converting units,
\(-3.00 \, \text{nC}\) becomes \(-3.00 \times 10^{-9} \, \text{C}\).

• Being negative, this charge will create an electric field that attracts positive charges towards it.
• The strength of this electric field can be calculated using Coulomb's Law,
• which helps in quantifying the effect of a point charge at various distances.

Electric Field Direction

The direction of the electric field is an essential concept when studying electric fields produced by point charges. Given a negative point charge, like the one in our exercise, the electric field lines point toward the charge.
This is because the electric field direction indicates how a positive test charge would move under the influence of the field.

For example, with our negative charge of \(-3.00 \, \text{nC}\), the electric field direction at any point around it is inward, toward the charge.

• This contrasts with a positive point charge, where the field would radiate outward.
• Being able to predict and determine these directions is vital for understanding interactions in electrostatic environments.

Knowing the field direction plays a crucial role in applications, from calculating forces exerted on other charges to understanding electric field distribution in space.