A point charge is placed at each corner of a square with side length aa. All charges have magnitude qq. Two of the charges are positive and two are negative (Fig. E21.4221.42). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of qq and aa?

Ch 21: Electric Charge and Electric Field

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 21: Electric Charge and Electric Field

Problem 42

Chapter 21, Problem 42

A point charge is placed at each corner of a square with side length $a$ . All charges have magnitude $q$ . Two of the charges are positive and two are negative (Fig. E $21.42$ ). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of $q$ and $a$ ?

Verified step by step guidance

First, identify the configuration of the charges. The square has four corners, and each corner has a point charge. Two charges are positive (+q) and two are negative (-q). Arrange them such that opposite corners have the same type of charge.

Next, consider the symmetry of the problem. The center of the square is equidistant from all four charges. Due to symmetry, the electric field contributions from opposite charges will have components that cancel each other out.

Calculate the electric field due to a single charge at the center of the square. The distance from the center to any corner is $ \frac{a}{\sqrt{2}} $. Use the formula for the electric field due to a point charge: $ E = \frac{k \cdot q}{r^2} $, where $ k $ is Coulomb's constant and $ r $ is the distance from the charge to the point of interest.

Determine the direction of the electric field vectors. The electric field due to a positive charge points away from the charge, while the field due to a negative charge points towards the charge. Analyze the vector components along the x and y axes for each charge.

Finally, sum the vector components of the electric fields from all four charges to find the net electric field at the center. Due to symmetry, the net electric field will have a specific direction along one of the diagonals of the square. Calculate the magnitude using vector addition of the individual electric fields.

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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 is a vector field around a charged particle that represents the force exerted per unit charge at any point in space. It is defined as E = F/q, where F is the force experienced by a test charge q. The direction of the electric field is the direction of the force on a positive test charge.

Superposition Principle

The superposition principle states that the net electric field due to multiple charges is the vector sum of the electric fields produced by each charge individually. This principle allows us to calculate the resultant field at a point by adding the contributions from each charge, considering both magnitude and direction.

Symmetry in Electric Fields

Symmetry plays a crucial role in simplifying electric field calculations. In this problem, the square configuration and equal magnitude of charges create symmetrical electric field contributions. This symmetry can be used to determine the direction and simplify the calculation of the net electric field at the center of the square.

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Textbook Question

Two positive point charges $q$ are placed on the $x$ -axis, one at $x = a$ and one at $x = -a$ . Derive an expression for the electric field at points on the $x$ -axis. Use your result to graph the $x$ -component of the electric field as a function of $x$ , for values of $x$ between $negative 4 a$ and $positive 4 a$ .

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Textbook Question

A point charge $q_{1} = - 4.00$ nC is at the point $x equals 0.600$ m, $y equals 0.800$ m, and a second point charge $q sub 2 equals positive 6.00$ nC is at the point $x equals 0.600$ m, $y equals 0$ . Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

Textbook Question

A $negative 4.00$ -nC point charge is at the origin, and a second $negative 5.00$ -nC point charge is on the $x$ -axis at $x equals 0.800$ m. Find the electric field (magnitude and direction) at each of the following points on the $x$ -axis: (i) $x equals 0.200$ m; (ii) $x equals 1.20$ m; (iii) $x = - 0.200$ m.

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Textbook Question

A very long, straight wire has charge per unit length $3.20 \times 10^{- 10}$ C/m. At what distance from the wire is the electric field magnitude equal to $2.50$ N/C?

Textbook Question

A charge of $negative 6.50$ nC is spread uniformly over the surface of one face of a nonconducting disk of radius $1.25$ cm. Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields? Note: Part (a) asked to find the magnitude and direction of the electric field this disk produces at a point $P$ on the axis of the disk a distance of $2.00$ cm from its center. Part (b) asked to find the magnitude and direction of the electric field at point $P$ , supposing that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Part (c) asked to find the magnitude and direction of the electric field at point $P$ if the charge is all brought to the center of the disk.

Textbook Question

A $negative 4.00$ -nC point charge is at the origin, and a second $-5.00$ -nC point charge is on the $x$ -axis at $x = 0.800$ m. Find the net electric force that the two charges would exert on an electron placed at each point in part (a). Note: Part (a) asked to find the electric field (magnitude and direction) at each of the following points on the $x$ -axis: (i) $x = 0.200$ m; (ii) $x = 1.20$ m; (iii) $x = -0.200$ m.

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