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Chapter 17: Problem 78

In a follow-up experiment, a charge of \(+40 \mathrm{pC}\) was placed at the center of an artificial flower at the end of a \(30-\mathrm{cm}\) -long stem. Bees were observed to approach no closer than \(15 \mathrm{~cm}\) from the center of this flower before they flew away. This observation suggests that the smallest external electric field to which bees may be sensitive is closest to which of these values? A. \(2.4 \mathrm{~N} / \mathrm{C}\) B. \(16 \mathrm{~N} / \mathrm{C}\) C. \(2.7 \times 10^{-10} \mathrm{~N} / \mathrm{C}\) D. \(4.8 \times 10^{-10} \mathrm{~N} / \mathrm{C}\)

Short Answer

Expert verified
The smallest external electric field is closest to 16 N/C (option B).

Step by step solution

01

Understanding the problem

We need to find the electric field at a distance of 15 cm from a point charge of +40 pC. The options provided are different values of the electric field.
02

Formula for electric field due to a point charge

The electric field \( E \) due to a point charge \( q \) at a distance \( r \) is given by the formula: \[ E = \frac{k \cdot q}{r^2} \] where \( k \) is the Coulomb's constant \( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \).
03

Substituting values into the formula

Given that \( q = 40 \, \text{pC} = 40 \times 10^{-12} \text{C} \) and \( r = 15 \text{ cm} = 0.15 \text{ m} \), plug these values into the formula: \[ E = \frac{8.99 \times 10^9 \cdot 40 \times 10^{-12}}{(0.15)^2} \]
04

Calculating the electric field

Calculate the expression from Step 3: \[ E = \frac{8.99 \times 10^9 \cdot 40 \times 10^{-12}}{0.0225} \approx \frac{359.6 \times 10^{-3}}{0.0225} \approx 16 \text{ N/C} \]
05

Concluding the answer

Based on our calculation, the electric field at 15 cm from the charge is approximately 16 N/C, which matches option B.

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Key Concepts

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

Coulomb's Law
Coulomb's Law is a fundamental principle in physics that describes the force between two point charges. This law states that the electric force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Mathematically, Coulomb's Law is expressed as:\[F = \frac{k \cdot |q_1 \cdot q_2|}{r^2}\]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 separating the charges.
  • \( k \) is Coulomb's constant \(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/ ext{C}^2\).
In the exercise, we use Coulomb's Law to determine how strong the electric field is at a given distance from a charge. It helps highlight the sensitivity of some creatures, like bees, to electric fields generated by such charges.
Point Charge
A point charge is an idealized model of a charge that assumes charge is concentrated at a single point in space. It simplifies calculations for electric fields and forces because interactions depend on the distance from this point. This is key when calculating the electric field produced by small charges as it minimizes extraneous variables.

In our exercise, we consider a point charge of \(+40 \, \text{pC}\). This small magnitude charge placed at the end of a flower's stem creates an electric field that bees can detect. This makes point charges ideal for theoretical exercises where physical size can be ignored while focusing on charges' effects at various distances.
Electric Field Sensitivity
Electric field sensitivity refers to the ability of organisms or devices to detect and respond to electric fields. In the given problem, bees demonstrated sensitivity to the electric field created by a point charge. They could sense the field strength and adjusted their behavior by flying away when it exceeded a certain threshold.

This threshold sensitivity implies that bees can perceive electric fields as low as those calculated in the problem (16 N/C). Understanding this sensitivity is crucial in understanding animal behavior and can also have applications in designing sensors for different technologies.
Physics Problem Solving
Problem solving in physics often involves understanding fundamental concepts and applying mathematical principles to reach a solution. The given exercise is a classic example where we first interpret the problem, identify the known values, and choose the correct formula, specifically for an electric field.

  • Begin by understanding the problem: identify what you're solving for and the given data. Here, the problem involves finding the electric field at a specific distance from a charge.
  • Select the correct formula: use knowledge of concepts like Coulomb's Law to find the formula that relates the given values to the unknown.
  • Substitute the known quantities into the formula: correctly input values, ensuring they are in compatible units (e.g., convert cm to m).
  • Calculate: carry out calculations methodically, checking each step for accuracy.
  • Review the result: consider whether it makes sense in context, as seen when matching calculated field strength to problem options.
Each of these steps builds your understanding and accuracy in dealing with physics questions, providing a framework for similar future problems.

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Most popular questions from this chapter

A proton is traveling horizontally to the right at \(4.50 \times 10^{6} \mathrm{~m} / \mathrm{s}\). (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of \(3.20 \mathrm{~cm}\). (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

In a rectangular coordinate system, a positive point charge \(q=6.00 \mathrm{nC}\) is placed at the point \(x=+0.150 \mathrm{~m}, y=0,\) and an identical point charge is placed at \(x=-0.150 \mathrm{~m}, y=0 .\) Find the \(x\) and \(y\) components and the magnitude and direction of the electric field at the following points: (a) the origin; (b) \(x=0.300 \mathrm{~m}, y=0 ;\) (c) \(x=0.150 \mathrm{~m}, y=-0.400 \mathrm{~m} ;\) (d) \(x=0, y=0.200 \mathrm{~m}\)

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

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 electrons. What strength of electric field does this proton produce at the distance of the electrons, approximately \(5.0 \times 10^{-10} \mathrm{~m}\) away?

Two equal point charges of \(+3.00 \times 10^{-6} \mathrm{C}\) are placed \(0.200 \mathrm{~m}\) apart. What are the magnitude and direction of the force each charge exerts on the other?
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