A charge is located at and a charge is located at . Thus the separation between the two charges is .
The total electric field on the axis can be written as
(a) What is the direction of the total electric field at any point on the axis where ?
(b)What is as a function of for ? Express your answer in terms of, if necessary, , , and the constant (if needed, enter pi for , epsilon_0 for ).
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(c) Consider now the limit where , so that
Express, in this limit, in terms of, if necessary, , and . The quantity is called the dipole moment.
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(a) To determine the direction of the total electric field at any point on the x-axis where y = 0, we can use the principle of superposition. The electric field due to each charge individually can be found using Coulomb's law, which states that the magnitude of the electric field due to a point charge is given by:
E = k * q / r^2
where:
- E is the electric field
- k is Coulomb's constant (approximately 9 x 10^9 Nm^2/C^2)
- q is the charge
- r is the distance between the charge and the point where the electric field is being calculated
In this case, we have two charges:
Charge 1 at position (x1, y1) = (0, d/2) with charge q1
Charge 2 at position (x2, y2) = (0, -d/2) with charge q2
The direction of the electric field due to each charge will be radially outward from the charge if it is positive, and radially inward towards the charge if it is negative.
To find the total electric field, we need to add the contributions of the individual electric fields due to each charge.
(b) To find the value of E as a function of x for y = 0, we need to calculate the magnitudes of the individual electric fields due to each charge and then sum them together.
Charge 1:
The distance, r1, between Charge 1 and the point on the x-axis is given by:
r1 = sqrt(x^2 + (d/2)^2)
The magnitude of the electric field due to Charge 1 is given by:
E1 = k * q1 / r1^2
Charge 2:
The distance, r2, between Charge 2 and the point on the x-axis is also given by:
r2 = sqrt(x^2 + (d/2)^2)
The magnitude of the electric field due to Charge 2 is given by:
E2 = k * q2 / r2^2
Then, the total electric field at any point on the x-axis can be written as:
E = E1 + E2
You can express your answer in terms of any necessary quantities, such as d, q1, q2, k, etc.
(c) In the limit where d approaches 0, the separation between the charges becomes infinitesimally small. This is known as the limit of an electric dipole.
In this case, the dipole moment, p, is given by the product of the charge magnitude, q1 or q2, and the separation distance, d:
p = q1 * d or p = q2 * d
In this limit, the electric field due to a dipole on the x-axis at any point can be approximated by:
E = (k * p) / x^3
where x is the distance from the center of the dipole on the x-axis.
You can express your answer in terms of necessary quantities, such as p, k, and x.