Can someone be kind enough to walk me through how you calculate an electric field at a point in three dimensions, due to two separate charges in the three dimensions.
The main part I am finding hard is the vector resolution.
Thank you so much, I can reward with lots of gold stars
The main part I am finding hard is the vector resolution.
Thank you so much, I can reward with lots of gold stars
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It depends what information you start with and in what form you want the answer. If the charges and the point are in a simple plane (e.g all in the x-y plane), the problem is simpler and I assume you know how to do that by finding the 2 fields and doing a vector addition. Here's a generalisation of the method in 3D.
q1 at (x1, y1, z1)
q2 at (x2, y2, z2)
Point, P, is (X, Y, Z)
For q1:
Distance from q1 to P, d1 = √[(X-x1)² + (Y-y1)² + (Z-z1)²]
Magnitude of field, E1, at P is:
|E1| = k.q1/d1²
Direction cosines of the vector E1 are:
x-axis: (X-x1)/d1
y-axis: (Y-y1)/d1
z-axis: (Z-z1)/d1
So the the electric field, E1, can be expressed in unit vector form, for example:
E1 = |E1|[((X-x1)/d1)i + ((Y-y1)/d1)j + ((Z-z1)/d1)k]
Similarly E2 can be expressed in component form:
E2 = |E1|[((X-x2)/d2)i + ((Y-y2)/d2)j + ((Z-z2)/d12)k]
The components can be added, e.g. If Ex is the total field in the x-direction at P:
Ex = |E1|((X-x1)/d1)+ |E2|((X-x2)/d2)
The total field E is therefore:
E= [|E1|((X-x1)/d1)+ |E2|((X-x2)/d2))]i
+ [(|E1|((Y-y1)/d1)+ |E2|((Y-y2)/d2))]j
+ [(|E1|((Z-z1)/d1)+ |E2|((Z-z2)/d2))]k
If required the magnitude and the direction cosines can then be found in the normal way:
|E| = √(Ex² + Ey² + Ez²)
Direction cosine for E with respect to x-axis is Ex/|E|
Direction cosine for E with respect to y-axis is Ey/|E|
Direction cosine for E with respect to z-axis is Ez/|E|
q1 at (x1, y1, z1)
q2 at (x2, y2, z2)
Point, P, is (X, Y, Z)
For q1:
Distance from q1 to P, d1 = √[(X-x1)² + (Y-y1)² + (Z-z1)²]
Magnitude of field, E1, at P is:
|E1| = k.q1/d1²
Direction cosines of the vector E1 are:
x-axis: (X-x1)/d1
y-axis: (Y-y1)/d1
z-axis: (Z-z1)/d1
So the the electric field, E1, can be expressed in unit vector form, for example:
E1 = |E1|[((X-x1)/d1)i + ((Y-y1)/d1)j + ((Z-z1)/d1)k]
Similarly E2 can be expressed in component form:
E2 = |E1|[((X-x2)/d2)i + ((Y-y2)/d2)j + ((Z-z2)/d12)k]
The components can be added, e.g. If Ex is the total field in the x-direction at P:
Ex = |E1|((X-x1)/d1)+ |E2|((X-x2)/d2)
The total field E is therefore:
E= [|E1|((X-x1)/d1)+ |E2|((X-x2)/d2))]i
+ [(|E1|((Y-y1)/d1)+ |E2|((Y-y2)/d2))]j
+ [(|E1|((Z-z1)/d1)+ |E2|((Z-z2)/d2))]k
If required the magnitude and the direction cosines can then be found in the normal way:
|E| = √(Ex² + Ey² + Ez²)
Direction cosine for E with respect to x-axis is Ex/|E|
Direction cosine for E with respect to y-axis is Ey/|E|
Direction cosine for E with respect to z-axis is Ez/|E|