Using the principle of superposition, we can calculate the electric field at point P on the x-axis by summing the electric fields created by each individual charge.
The electric field created by a point charge is given by Coulomb's law:
E = k*q/r^2
where E is the electric field, k is Coulomb's constant, q is the charge, and r is the distance from the charge to the point where we want to calculate the electric field.
Let's consider the electric field created by the charge at y = +d. The distance between this charge and point P is given by r1 = sqrt(x^2 + d^2). The magnitude of the electric field created by this charge is then:
E1 = k*(-q)/r1^2 = -k*q/(x^2 + d^2)
Similarly, the electric field created by the charge at y = -d is given by:
E2 = k*(-q)/(x^2 + d^2)
Finally, the electric field created by the charge at y = 0 is given by:
E3 = k*(2q)/(x^2 + 0^2) = 2k*q/x^2
The total electric field at point P is the sum of these individual electric fields:
E_total = E1 + E2 + E3
= -k*q/(x^2 + d^2) + k*(-q)/(x^2 + d^2) + 2k*q/x^2
= -k*q*(1/(x^2 + d^2) - 1/x^2) + 2k*q/x^2
= -k*q/x^2 + k*q/(x^2 + d^2) + 2k*q/x^2
= k*q/(x^2 + d^2)
Therefore, the expression for the electric field at point P on the x-axis is:
E_total = k*q/(x^2 + d^2)
Three charges are on the y axis. Two of the charges, each --q, are located y = ±d, and the third charge, +2q, is located at y = 0. Derive an expression for the electric field at a point P on the x-axis.
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