Calculating Electric Field Intensity at (0, 0, z)

[technicolour1]

Homework Statement

Electric charge of density ρℓ = 5 μC/m is distributed along the arc r = 2 cm,
0 ≤ ∅ ≤ ∏/4, z = 0. Find the electric field intensity at (0, 0, z)

Homework Equations

dE=(dQ(r)/(4∏ε(r^2))) (unit vector R)

The Attempt at a Solution

Letting pL be the charge density, letting r^2 = mag(r)^3

pLdL = dQ
dL = rd∅
pLrd∅ = dQ, r=0.02

Note: Limits of integration are 0 -> ∏/4

E(r) = 5E-6*0.02/(4∏ε)∫(z*(unitvector z) - 0.02cos∅* (unit vector x) - 0.02sin∅*(unit vector y))/((z^2+0.0004)^(3/2))) d∅

=900∫(z*(unitvector z) - 0.02cos∅* (unit vector x) - 0.02sin∅*(unit vector y))/((z^2+0.0004)^(3/2))) d∅I found it kind of hard to use notation while typing, so feel free to ask me to clarify. I know what I just typed is extremely tangly, so thanks in advance for taking the time to look at it.

My question is, did I go about this problem the right way? I don't feel confident about the evaluation I did to get all variables in terms of ∅. Also, I brought the unit vector into the calculation so that the final answer's vector will depend on what's being plugged in. I figure since z must be a constant point charge, the integral can be easily evaluated once a z value is given.

 

[SammyS]

Homework Statement

Electric charge of density ρℓ = 5 μC/m is distributed along the arc r = 2 cm,
0 ≤ ∅ ≤ ∏/4, z = 0. Find the electric field intensity at (0, 0, z)

Homework Equations

dE=(dQ(r)/(4∏ε(r^2))) (unit vector R)

The Attempt at a Solution

Letting pL be the charge density, letting r^2 = mag(r)^3

pLdL = dQ
dL = rd∅
pLrd∅ = dQ, r=0.02

Note: Limits of integration are 0 -> ∏/4

E(r) = 5E-6*0.02/(4∏ε)∫(z*(unitvector z) - 0.02cos∅* (unit vector x) - 0.02sin∅*(unit vector y))/((z^2+0.0004)^(3/2))) d∅

=900∫(z*(unitvector z) - 0.02cos∅* (unit vector x) - 0.02sin∅*(unit vector y))/((z^2+0.0004)^(3/2))) d∅I found it kind of hard to use notation while typing, so feel free to ask me to clarify. I know what I just typed is extremely tangly, so thanks in advance for taking the time to look at it.

My question is, did I go about this problem the right way? I don't feel confident about the evaluation I did to get all variables in terms of ∅. Also, I brought the unit vector into the calculation so that the final answer's vector will depend on what's being plugged in. I figure since z must be a constant point charge, the integral can be easily evaluated once a z value is given.

The result looks correct.

You're right. It is rather hard to follow.