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24. Electric Force & Field; Gauss' Law

Electric Field

Physics

24. Electric Force & Field; Gauss' Law

Electric Field

Previous problemNext problem

Problem 15a

Textbook Question

Two 10-cm-diameter charged rings face each other, 20 cm apart. The left ring is charged to −20 nC and the right ring is charged to +20 nC. What is the electric field Ē, both magnitude and direction, at the midpoint between the two rings?

Verified step by step guidance

Determine the position of the midpoint between the two rings. Since the rings are 20 cm apart, the midpoint is located 10 cm from each ring along the axis connecting their centers.

Recall the formula for the electric field along the axis of a charged ring:

E = \frac{k Q z}{(\sqrt{R^{2} + z^{2}}) 3}

, where

k

is Coulomb's constant,

Q

is the charge on the ring,

R

is the radius of the ring, and

z

is the distance from the center of the ring to the point where the field is being calculated.

Substitute the given values into the formula for each ring. For the left ring,

Q = -20 nC

R = 5 cm

, and

z = 10 cm

. For the right ring,

Q = +20 nC

R = 5 cm

, and

z = 10 cm

. Note that the electric field due to each ring will point in opposite directions because the charges are opposite.

Calculate the magnitude of the electric field produced by each ring at the midpoint using the formula. Since the midpoint is equidistant from both rings and the charges are equal in magnitude but opposite in sign, the magnitudes of the electric fields will be the same, but their directions will differ.

Combine the electric fields from both rings. Because the fields point in opposite directions, their magnitudes will add. Determine the net electric field at the midpoint, and specify its direction (toward the positively charged ring).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field

The electric field (E) is a vector field that represents the force per unit charge exerted on a positive test charge placed in the field. It is defined as E = F/q, where F is the force experienced by the charge and q is the magnitude of the charge. The direction of the electric field is away from positive charges and towards negative charges.

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 individually. This principle allows us to calculate the net electric field at a point by considering the contributions from each charge separately, taking into account their magnitudes and directions.

Distance and Charge Relationship

The strength of the electric field produced by a point charge decreases with the square of the distance from the charge, as described by Coulomb's law. For a charged ring, the electric field at a point along its axis can be calculated using the charge and the distance from the ring, which is crucial for determining the electric field at the midpoint between the two rings in this scenario.

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Related Practice

Textbook Question

The electric field in a region of space is Ex = −1000x^2 V/m, where x is in meters. Graph Ex versus x over the region −1 m ≤ x ≤1 m.

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Textbook Question

An electric dipole at the origin consists of two charges ±q spaced distance s apart along the y-axis. What is the field Ē on the bisecting axis? Does your result agree with Equation 23.11?

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Textbook Question

An infinitely long cylinder of radius R has linear charge density λ. The potential on the surface of the cylinder is V₀, and the electric field outside the cylinder is E_r = λ/2πϵ₀r . Find the potential relative to the surface at a point that is distance r from the axis, assuming r>R.

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Textbook Question

FIGURE P23.44 shows a thin rod of length L with total charge Q. Evaluate E at r=3.0 cm if L=5.0 cm and Q=3.0 nC.

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Textbook Question

A ring of radius R has total charge Q. At what distance along the z-axis is the electric field strength a maximum?

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Textbook Question

A rod of length L lies along the y-axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Find the electric field strength of the rod at distance x on the x-axis.

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Textbook Question

An infinite plane of charge with surface charge density 3.2 μC/m² has a 20-cm-diameter circular hole cut out of it. What is the electric field strength directly over the center of the hole at a distance of 12 cm? Hint: Can you create this charge distribution as a superposition of charge distributions for which you know the electric field?

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Multiple Choice

A 1.5μC charge, with a mass of 50g, is in the presence of an electric field that perfectly balances its gravity. What magnitude does the electric field need to be, and in what direction does it need to point?

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