Concept explainers
Four equal-magnitude (4.0
(a)
The electric field at the center of the square, iffour positively charged charges, having magnitude of each is
Answer to Problem 42SP
Solution:
Explanation of Solution
Given data:
The length of each side of the square is
The magnitude of each charge is
Formula used:
Write the expression for the electric field
Here,
Explanation:
In this case, all charges are of equal magnitude and polarity, and are located at the same distance from the center of the square.
Denote the charge by
As all charges are positive, the direction of the electric field shall be outward and away from the charges, as shown in the figure below.
The electric field at the center of the square will the resultant electric filed due to all the electric fields acting on it. Therefore, the electric field at the center of the square will be the resultant of the electric fields due to charges at points A, B, C, and D.
Recall the expression for the electric field.
Refer to the above diagram and write the expression for the electric field at the corners of the square due to all the four-point charges that are placed at the corners of the square.
Write the expression for the electric field due to charges at the point A.
Write the expression for the electric field due to charges at the point B.
Write the expression for the electric field due to charges at the point C.
Write the expression for the electric field due to charges at the point D.
The resultant electric field at the center of the square will be
Substitute
Conclusion:
The electric field at the center of the square if the charges are all positive is
(b)
The electric field at the center of the square if the charges alternate in sign around the perimeter of the square and the magnitude of each charge is
Answer to Problem 42SP
Solution:
Explanation of Solution
Given data:
The length of each side of the square is
The magnitude of each charge is
Formula used:
Write the expression for the electric field
Here,
Explanation:
In this case, all charges are of equal magnitude, however, they are placed such that they alternate in sign around the perimeter of the square. Also, they are located at the same distance from the center of the square.
Denote the charge by
For positivecharges, the direction of the electric field shall be outward and away from the charges, while for negative charges the direction of the electric field shall be toward the charges as shown in the figure below.
The electric field at the center of the square will be the resultant electric filed due to all the electric fields acting on it. Therefore, the electric field at the center of the square will be the resultant of the electric fields due to charges at points A, B, C and D.
The expression for the electric field is given as
Here,
Write the expression for the electric field due to charges at the point A.
Write the expression for the electric field due to charges at the point B.
Write the expression for the electric field due to charges at the point C.
Write the expression for the electric field due to charges at the point D.
The resultant electric field at the center of the square will be
Substitute
Conclusion:
The electric field at the center of the square, if the charges alternate in sign around the perimeter of the square, is
(c)
The electric field at the center of the square if the all the four charges, having magnitude of each as
Answer to Problem 42SP
Solution:
Explanation of Solution
Given data:
The length of each side of the square is
The magnitude of each charge is
Formula used:
Write the expression for the electric field.
Here,
Write the expression for the relation between the length of the diagonal and side of a square
Here,
Write the expression for the net (resultant) electric field
Here,
Explanation:
In this case, all charges are of equal magnitude, however, they are placed such the charges on the top are positive while the charges on the bottom are negative. Also, they are located at the same distance from the center of the square.
Denote the charge by
For positive charges, the direction of the electric field shall be outward and away from the charges, while for negative charges the direction of the electric field shall be toward the charges, as shown in the figure below.
The electric field at the center of the square will be the resultant electric filed due to all the electric fields acting on it. Therefore, the electric field at the center of the sphere will be the resultant of the electric fields due to charges at points A, B, C and D.
Recall the expression for diagonal length of a square.
Here,
Apply the equation mentioned above to determine the distance
Write the expression for distance
Substitute
Write the expression for the electric field.
Here,
Refer to the above diagram and write the expression for the electric field due to charges at A, B, C and D.
Write the expression for the electric field due to charges at the point A.
Write the expression for the electric field due to charges at the point B.
Write the expression for the electric field due to charges at the point C.
Write the expression for the electric field due to charges at the point D.
Recall the expression for the resultant electric field at the center of the square.
The angle between sum of electric field due to C and A, and D and B is
Substitute
Understand that
Substitute
Conclusion:
The electric field at the center of the square if the charges follow the sequence plus, plus, minus, minus around the perimeter of the square is
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