A charge −Q is located at x=−l/2 and a charge +Q is located at x=l/2. Thus the separation between the two charges is l.
(b)What is E(x) as a function of x for x>l/2? Express your answer in terms of, if necessary, Q, l, x and the constant ϵ0 (if needed, enter pi for π, epsilon_0 for ϵ0).
(c) Consider now the limit where x≫l, so that
d)Express, in this limit, E(x) in terms of, if necessary, p, x and ϵ0. The quantity p=Ql is called the dipole moment.
To find the electric field E(x) for x > l/2, we need to consider the contributions from both charges at their respective positions.
For a point at x > l/2, the charge -Q is closer than the charge +Q, so the electric field from -Q will dominate.
(a) Electric field due to a point charge: The electric field at a distance r from a point charge Q is given by the equation:
E = k * Q / r²
where k is the electrostatic constant and equals 1 / (4 * pi * epsilon_0).
(b) Electric field due to the negative charge (-Q): The distance from the negative charge to the point at x is x - (-l/2) = x + l/2. Using the equation from (a), we can write:
E_negative = k * (-Q) / (x + l/2)²
(c) Electric field due to the positive charge (+Q): The distance from the positive charge to the point at x is x - (l/2) = x - l/2. Similarly, using the equation from (a), we have:
E_positive = k * (+Q) / (x - l/2)²
Since we are considering the limit where x ≫ l, we can neglect the constant term of l/2 in the denominators. Additionally, since the charges have equal magnitude (Q and -Q), the magnitudes of the electric fields are equal.
Therefore, in this limit, E(x) is approximately equal to E_negative:
E(x) ≈ E_negative
≈ k * (-Q) / (x + l/2)²
Substituting the value of k, we get:
E(x) ≈ -(1 / (4 * pi * epsilon_0)) * Q / (x + l/2)²
This expresses E(x) in terms of Q, l, x and the constant epsilon_0.
(d) In the limit where x ≫ l, we can also express the electric field E(x) in terms of the dipole moment p = Q * l.
Recall that p is the product of the charge magnitude Q and the separation distance l. Therefore, we can rewrite Q as p / l in the expression for E(x):
E(x) ≈ -(1 / (4 * pi * epsilon_0)) * (p / l) / (x + l/2)²
Simplifying further:
E(x) ≈ -(1 / (4 * pi * epsilon_0)) * p / (l * (x + l/2)²)
This gives the expression for E(x) in terms of p, x, and epsilon_0 in the limit where x ≫ l.