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Ch 26: Potential and Field
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 26: Potential and FieldProblema 49c
Capitolo 26, Problema 49c

Two positive point charges q are located on the y-axis at y = ±a. Symmetry dictates that the electric field along the x-axis has only an x-component: Ey=Ez=0. Find an expression for Ex if x≪a.

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Step 1: Begin by recalling Coulomb's law, which states that the electric field due to a point charge is given by \( E = \frac{kq}{r^2} \), where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the point of interest.
Step 2: Identify the geometry of the problem. The two charges \( q \) are located at \( y = +a \) and \( y = -a \), and the point of interest is along the x-axis at a distance \( x \) from the origin. The distance \( r \) from each charge to the point on the x-axis can be expressed as \( r = \sqrt{x^2 + a^2} \).
Step 3: Decompose the electric field contributions from each charge into components. The electric field due to each charge has both an x-component and a y-component. By symmetry, the y-components cancel out, leaving only the x-components. The x-component of the electric field from each charge is \( E_x = E \cdot \frac{x}{r} \), where \( E \) is the magnitude of the electric field.
Step 4: Sum the x-components of the electric field from both charges. Since the charges are identical and symmetrically placed, the total x-component of the electric field is \( E_x = 2 \cdot \frac{kq}{r^2} \cdot \frac{x}{r} \). Substitute \( r = \sqrt{x^2 + a^2} \) into the expression.
Step 5: Simplify the expression for \( E_x \) under the condition \( x \ll a \). When \( x \ll a \), \( \sqrt{x^2 + a^2} \approx a \), and \( r^2 \approx a^2 \). Using this approximation, the expression for \( E_x \) simplifies to \( E_x \approx \frac{2kxq}{a^3} \).

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Electric Field

The electric field is a vector field that represents the force exerted by electric charges on other charges in space. It is defined as the force per unit charge and is directed away from positive charges and towards negative charges. The strength and direction of the electric field depend on the magnitude and position of the source charges.
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Intro to Electric Fields

Superposition Principle

The superposition principle states that the total electric field created by multiple point charges is the vector sum of the electric fields produced by each charge individually. This principle allows us to analyze complex charge configurations by considering the contributions from each charge separately and then combining them to find the resultant field.
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Superposition of Sinusoidal Wave Functions

Symmetry in Electric Fields

Symmetry in electric fields refers to the predictable patterns that arise from the arrangement of charges. In this case, the two positive charges are symmetrically placed along the y-axis, leading to the conclusion that the electric field along the x-axis will have no y or z components (Ey = Ez = 0). This simplifies the analysis, allowing us to focus solely on the x-component of the electric field.
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Electric Field Lines
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