Three charges, +2.5 micro coulomb, -4.8 micro coulomb and 6.3 micro coulomb, are located at (0.20m, .15m), (.50m, -.35m), and (-.42m,-.32m) respectively. What is the electric field at the origin?
To do this you need to take each charge one at a time. Use E = kq/r^2 for each to get the magnitude (you'll have to do a little pythagorean for each to get r) and the direction will be either towards or away from the origin along the line formed by the point and the origin. Frankly it's probably easier to do a tan-1 and find the angle, then do sin and cos to find the components. Sum the three x and y components for the final vector.
For example for the first
E = 9e9 * 2.5e-6 / .25^2 at an angle of tan-1(.15/.2) = 36.9o away from the origin. Find x, y components. Do same for the other two.
To find the electric field at the origin, we need to calculate the electric field contribution from each charge and then sum them up vectorially.
The electric field due to a point charge at a given location can be calculated using Coulomb's Law:
E = k * (Q / r^2) * u
where E is the electric field, k is the electrostatic constant (9.0 x 10^9 N m^2/C^2), Q is the charge, r is the distance between the charge and the point of interest, and u is the unit vector pointing from the charge to the point of interest.
Let's calculate the electric field due to each charge:
Charge 1: +2.5 μC at (0.20 m, 0.15 m)
- The distance from charge 1 to the origin is:
r1 = sqrt((0.20 m)^2 + (0.15 m)^2) = 0.25 m
- The unit vector from charge 1 to the origin is:
u1 = (0.20 m / r1, 0.15 m / r1) = (0.80, 0.60)
- The electric field due to charge 1 is:
E1 = k * (2.5 x 10^-6 C / (0.25 m)^2) * (0.80, 0.60) = (k * 8.0 x 10^-2) * (0.80, 0.60)
Charge 2: -4.8 μC at (0.50 m, -0.35 m)
- The distance from charge 2 to the origin is:
r2 = sqrt((0.50 m)^2 + (-0.35 m)^2) = 0.61 m
- The unit vector from charge 2 to the origin is:
u2 = (0.50 m / r2, -0.35 m / r2) = (0.82, -0.57)
- The electric field due to charge 2 is:
E2 = k * (-4.8 x 10^-6 C / (0.61 m)^2) * (0.82, -0.57) = (k * -13.06) * (0.82, -0.57)
Charge 3: 6.3 μC at (-0.42 m, -0.32 m)
- The distance from charge 3 to the origin is:
r3 = sqrt((-0.42 m)^2 + (-0.32 m)^2) = 0.53 m
- The unit vector from charge 3 to the origin is:
u3 = (-0.42 m / r3, -0.32 m / r3) = (-0.79, -0.61)
- The electric field due to charge 3 is:
E3 = k * (6.3 x 10^-6 C / (0.53 m)^2) * (-0.79, -0.61) = (k * 22.8) * (-0.79, -0.61)
Now, let's sum up the contributions from each charge vectorially:
E_total = E1 + E2 + E3
By plugging in the values for k and calculating the vector sum of E_total, we can find the electric field at the origin.