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PROBLEMS 1. A field is given in spherical coordinates at P(r = 5, θ = 30°,φ= 60°) as E = 20a, -30a_θ, +60aφ, V/m. Find the incremental work done in moving a 10-μC charge a distance of 0. 8 μm in the direction: (a) a,; (b) a_φ; (c) a_φ; (d) of E; (e) of G = 2a_x + 4a_y– 3a_x

User Everettss
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Answer:

To find the incremental work done, we can use the formula:

dW = F · dr,

where F is the force and dr is the displacement vector.

(a) To find the incremental work in the direction of a:

The force in the direction of a can be found by taking the dot product of the field E and the unit vector a:

F = E · a = (20a) · a = 20 |a|^2 = 20,

since |a| = 1 (unit vector).

The displacement vector dr is given by dr = a dr, where dr = 0.8 μm.

Therefore, the incremental work is:

dW = F · dr = 20 · (a · a) dr = 20 dr = 20(0.8 μm) = 16 μJ.

(b) To find the incremental work in the direction of a_φ:

The force in the direction of a_φ can be found by taking the dot product of the field E and the unit vector a_φ:

F = E · a_φ = (20a) · (-30a_θ) = 0,

since a · a_φ = 0 (perpendicular vectors).

Therefore, the incremental work is:

dW = F · dr = 0.

(c) To find the incremental work in the direction of a_φ:

The force in the direction of a_φ can be found by taking the dot product of the field E and the unit vector a_φ:

F = E · a_θ = (20a) · (-30a_θ) = 0,

since a · a_θ = 0 (perpendicular vectors).

Therefore, the incremental work is:

dW = F · dr = 0.

(d) To find the incremental work in the direction of E:

The force in the direction of E can be found by taking the dot product of the field E and the unit vector E:

F = E · E = (20a) · (20a) = 20^2 = 400,

since |E| = 20.

The displacement vector dr is given by dr = E dr, where dr = 0.8 μm.

Therefore, the incremental work is:

dW = F · dr = 400 dr = 400(0.8 μm) = 320 μJ.

(e) To find the incremental work in the direction of G = 2a_x + 4a_y - 3a_z:

The force in the direction of G can be found by taking the dot product of the field E and the unit vector G:

F = E · G = (20a) · (2a_x + 4a_y - 3a_z) = 40a · a_x + 80a · a_y - 60a · a_z = 40,

since a · a_x = 1, a · a_y = 0, and a · a_z = 0.

The displacement vector dr is given by dr = G dr, where dr = 0.8 μm.

Therefore, the incremental work is:

dW = F · dr = 40 dr = 40(0.8 μm) = 32 μJ.

So, the incremental work done in each direction is:

(a) 16 μJ

(b) 0

(c) 0

(d) 320 μJ

(e) 32 μJ.

User Gaurav S
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