Ch. 21 - Electric Charge and Electric Field

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 21 - Electric Charge and Electric Field

문제 40

21장, 문제 40

You are given two unknown point charges, Q₁ and Q₂. At a point on the line joining them, one-third of the way from Q₁ to Q₂, the electric field is zero (Fig. 21–64). What is the ratio Q₁/Q₂?

검증된 단계별 안내

Identify the key concept: The electric field at a point due to a point charge is given by the formula:

E = \frac{k Q}{r^{2}}

, where

k

is Coulomb's constant,

Q

is the charge, and

r

is the distance from the charge to the point of interest. The electric field is zero at the given point, meaning the fields due to

Q_{1}

and

Q_{2}

must cancel each other out.

Define the distances: Let the total distance between

Q_{1}

and

Q_{2}

d

. The point where the electric field is zero is located one-third of the way from

Q_{1}

Q_{2}

. Therefore, the distance from

Q_{1}

to the point is

\frac{d}{3}

, and the distance from

Q_{2}

to the point is

\frac{2}{3} d

Set up the condition for the electric field to be zero: The magnitudes of the electric fields due to

Q_{1}

and

Q_{2}

must be equal at the point. Using the formula for electric field, this gives:

\frac{k Q_{1}}{{\frac{d}{3}}^{2}} = \frac{k Q_{2}}{{\frac{2}{3}}^{2}}

Simplify the equation: Cancel out

k

on both sides and rearrange the terms to isolate the ratio

\frac{Q_{1}}{Q_{2}}

. This gives:

\frac{Q_{1}}{Q_{2}} = \frac{{\frac{2}{3}}^{2}}{{\frac{1}{3}}^{2}}

Evaluate the ratio: Simplify the expression for the ratio

\frac{Q_{1}}{Q_{2}}

by calculating the squares of the fractions. This will yield the final ratio of

\frac{Q_{1}}{Q_{2}}

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as the force per unit charge and is directed away from positive charges and toward negative charges. The strength of the electric field decreases with distance from the charge, following an inverse square law.

Superposition Principle

The superposition principle states that the total electric field created by multiple charges is the vector sum of the electric fields produced by each charge independently. This principle allows us to analyze complex charge configurations by considering the contributions of individual charges to the overall electric field at a given point.

Charge Ratio

The charge ratio, in this context, refers to the relationship between the magnitudes of two point charges, Q₁ and Q₂. To find the ratio Q₁/Q₂, one must consider the conditions under which the electric field is zero at a specific point, which involves balancing the electric fields due to both charges. This ratio is crucial for determining how the magnitudes of the charges influence the electric field at that point.