3 There are two charges +Q and –Q/2 that locate at two positions with a separation of d as shown in Fig. 1. We
may put +Q at the origin of the x-axis to facilitate the calculation.
(a) Find the electric field for +Q and –Q/2 at the position P. (5 points)
(b) What is the total electric field at P? The answer should include the magnitude and direction of the
electric filed. (5 points)
(c) At what position(s) x is the electric field zero? (5 points)
(d) Is the field zero at any other points, not on the x axis? (5 points)
LOL EE210?
2+2=4
4-1=3 Quick Maths!!!
To find the electric field at position P due to the charges +Q and -Q/2, we can use the principle of superposition. The electric field at P will be the vector sum of the individual electric fields due to each charge.
(a) Let's start by finding the electric field at P due to the charge +Q:
The electric field due to a point charge is given by Coulomb's law:
E = k * Q / r^2
Here, k is the Coulomb's constant, Q is the charge, and r is the distance between the charge and position P.
Since the charge +Q is located at the origin (x = 0), the distance between +Q and P is d.
So, the electric field at P due to +Q is:
E1 = k * Q / d^2
Next, let's find the electric field at P due to the charge -Q/2:
Since the charge -Q/2 is located at a distance d from +Q, the distance between -Q/2 and P is 2d.
So, the electric field at P due to -Q/2 is:
E2 = k * (-Q/2) / (2d)^2
(b) To find the total electric field at P, we need to sum the individual electric fields: E_total = E1 + E2.
The magnitudes of the electric fields E1 and E2 were calculated in part (a). The total electric field at P will be the vector sum of these magnitudes. To find the direction, we can use the principle of superposition by adding the vectors.
(c) To find the position(s) x where the electric field is zero, we need to set E_total = 0 and solve for x.
(d) The field may be zero at other points not on the x-axis, depending on the position and magnitude of the charges. However, without specific values for Q, d, and other details, we cannot determine these points.
To find the electric field at position P, we can use Coulomb's law, which states that the electric field created by a point charge is given by:
E = k * Q / r^2,
where E is the electric field, k is Coulomb's constant (9.0 x 10^9 N m^2/C^2), Q is the charge, and r is the distance from the charge to the point where we want to calculate the electric field.
a) The electric field at position P due to the +Q charge:
Since +Q is at the origin of the x-axis, the distance from +Q to P is simply the x-coordinate of P, which is d.
E_1 = k * Q / d^2.
b) The electric field at position P due to the -Q/2 charge:
The distance from -Q/2 to P is the distance between the two charges minus the x-coordinate of P. So, the distance is d - x.
E_2 = k * (-Q/2) / (d - x)^2.
c) To find the position(s) x where the electric field is zero, we equate E_1 and E_2 and solve for x:
k * Q / d^2 = k * (-Q/2) / (d - x)^2.
By cross-multiplying and simplifying, we get:
Q * (d - x)^2 = (-Q/2) * d^2.
Expanding (d - x)^2 and simplifying further, we obtain:
d^2 - 2dx + x^2 = -d^2/2.
Rearranging the terms, we get:
2x^2 - 2dx + (d^2 - d^2/2) = 0.
Simplifying, we find:
2x^2 - 2dx + d^2/2 = 0.
Solving this quadratic equation will give us the position(s) x where the electric field is zero.
d) To check if the electric field is zero at any other points not on the x-axis, we need to calculate the electric field at those points using the same formula as (a) and (b). If the resulting electric field is equal to zero, then the field is indeed zero at those points.