Two van de Graaff generators, whose centers are separated from one another by 0.50 m, become charged after they are switched on. One van de Graaff generatord holds +3.0 x10^2 C, while the other holds -2.0 x 10^2 C. What is the magnitude and direction of the electric field halfway between them?

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A van de Graaff generator can be thought of as a charged sphere. Thus, in this problem, we need to find the field halfway between the two spheres, one with the charge `q_1 = 3*10^2 C` and the other one with the charge `q_2 = -2*10^2 C` .

The electric...

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A van de Graaff generator can be thought of as a charged sphere. Thus, in this problem, we need to find the field halfway between the two spheres, one with the charge `q_1 = 3*10^2 C` and the other one with the charge `q_2 = -2*10^2 C` .

The electric field of a sphere is directed radially outward for a positively charged sphere and radially inward for a negatively charged sphere. (Please see the attached image for an illustration.) The magnitude of the field outside of a charged sphere does not depend on its radius, only on the distance between a given point and the center of the sphere.

Thus, halfway between the van de Graaff generators, the electric field of the first one is directed to the right (again, see the attached image) and its magnitude is

`E_1 = k|q_1|/r^2` , where k = `8.99*10^9 N*m^2/C^2` is a Coulomb's constant and r is half the distance between the centers of the generators: r = 0.25 m.

The electric field of the second one, at the same point, is also directed to the right and its magnitude is

`E_2 = k|q_2|/r^2` .

According to the superposition principle, the total field at this point is the vector sum of the fields due to the both generators. Since the field vectors have the same direction, the magnitude of the sum is the sum of their magnitudes:

`E = E_1 + E_2 = k(q_1+q_2)/r^2 = 7.2*10^13 N/C` .

So the electric field at the point halfway between the two van de Graaffs generators is directed along the line connecting their centers, towards the second generator, and its magnitude is 7.2*10^13 N/C.

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