Solving Two Point Charges: Electric Field, Dipole Moment & Potential Energy

[bmarson123]

Homework Statement

Two point charges q₁ = 2\muC and q₂ = -2\muC are placed at r₁ = (3,0,0) m and r₂ = (0,0,4)m respectively

i) What is the force of q₁ (in vector form)?

ii) What is the electric field at the origin?

iii) What is the electric dipole moment of this arrangement (in vector form)?

iv) What is the potential energy contained in this arrangement of charges?

Homework Equations

F =[ k q₁q₂ / (mod (r₁-r₂) )³ ] (r₁ - r₂)

E_T = \sum E_i

E = (mod q_i) / 4\pi\epsilon mod r_i

p = qd

U =Qq / 4\pi\epsilon mod (r₂ - r₁)

The Attempt at a Solution

i) F = k [ -4x10^-9 / 5³ (3,0,-4)
= 0.288 (3,0,-4) N

ii) E₁ = 2x10^-9 / 4\pi\epsilon3
= 5.99454 i

E₂ = 2x10^-9 / 4\pi\epsilon4
= 4.49590 k

E_T = (6.0,0,4.5) N/C

iii) d = r₂ - r₁
= (-3,0,4)

p =2x10^-9 (-3,0,4)
= (-6x10^-9 , 0, 8x10^-9) C.m

iv) U = -4x10^-9 /4\pi\epsilon 5 = 7.2 J

Basically, I just wanted to know if this is all right because it comes up on every single exam and I don't want to think it's right for it not to be!

 

[quietrain]

Homework Statement

Two point charges q₁ = 2\muC and q₂ = -2\muC are placed at r₁ = (3,0,0) m and r₂ = (0,0,4)m respectively

i) What is the force of q₁ (in vector form)?

ii) What is the electric field at the origin?

iii) What is the electric dipole moment of this arrangement (in vector form)?

iv) What is the potential energy contained in this arrangement of charges?

Homework Equations

F =[ k q₁q₂ / (mod (r₁-r₂) )³ ] (r₁ - r₂)

E_T = \sum E_i

E = (mod q_i) / 4\pi\epsilon mod r_i

p = qd

U =Qq / 4\pi\epsilon mod (r₂ - r₁)

The Attempt at a Solution

i) F = k [ -4x10^-9 / 5³ (3,0,-4)
= 0.288 (3,0,-4) N

ii) E₁ = 2x10^-9 / 4\pi\epsilon3
= 5.99454 i

E₂ = 2x10^-9 / 4\pi\epsilon4
= 4.49590 k

E_T = (6.0,0,4.5) N/C

iii) d = r₂ - r₁
= (-3,0,4)

p =2x10^-9 (-3,0,4)
= (-6x10^-9 , 0, 8x10^-9) C.m

iv) U = -4x10^-9 /4\pi\epsilon 5 = 7.2 J

Basically, I just wanted to know if this is all right because it comes up on every single exam and I don't want to think it's right for it not to be!

i) i assume question wants force OF q1 ON q2
your ans seems weird
microCoulomb is 10^-6 , so q1q2 = -4 x 10^-12

also, your formula of (r₁-r₂) means r₂₁ = force of q2 TO q1. which is not the question. the question wants force of q1 TO q2. so your formula should read r₁₂ = r₂-r₁

doing so gives you r₁₂= (-3,0,4)

now -ve q and +ve q gives you attraction = -q₁q₂

so when you put back into forumla, you get -r₁₂=r₂₁

thats how you get force of Q1 ON Q2, which is attractive, means it pulls Q2 towards Q1, given by r₂₁

if you are still confused, read this , scroll down to vector form
http://en.wikipedia.org/wiki/Coulomb's_law

 

[quietrain]

ii) the electric field at a point is the superposition of the point charges everywhere else

so question wants at origin.

at origin, the E-field from +Q1 is towards the -ve x direction.
at origin, the E-field from -Q2 is towards the +ve z direction.

so your resultant direction is not (6,0,4.5), i think your magnitude is wrong too

E = kQ₁ r₁ / r₁³ + -kQ₂r₂ /r₂³

so k = 9x10⁹
Q₁ = +2x10^-6
Q₂ = -2x10^-6
r₁³ = 3³=27
r₁=(-3,0,0)
r₂³ = 4³=64
r₂=(0,0,4)

resultant direction = r₁+r₂=(-3,0,4)
magnitude = (666, 0 , 281)
E-field at origin = (-3*666, 0 , 4*281) N/C

 

[quietrain]

iii) the electric dipole's displace vector d points from -ve to +ve charge.

so d is not (-3,0,4) but rather (3,0,-4)

your q should be 2*10^-6

 

[quietrain]

iv) potential energy = kQ1Q2/ (distance between charges = 5) , again your magnitude is incorrect

 

[bmarson123]

Thanks so much for your help!

I'm still a bit confused about i)

I thought that if it wants the force on q₁ then surely it should be the force of q₂ on q₁? Because otherwise it'll be the force q₁ is excerting on q₂?

I did look at the link but it just confused me more, could you explain it a bit more for me?

 

[bmarson123]

ii) the electric field at a point is the superposition of the point charges everywhere else

so question wants at origin.

at origin, the E-field from +Q1 is towards the -ve x direction.
at origin, the E-field from -Q2 is towards the +ve z direction.

so your resultant direction is not (6,0,4.5), i think your magnitude is wrong too

E = kQ₁ r₁ / r₁³ + -kQ₂r₂ /r₂³

so k = 9x10⁹
Q₁ = +2x10^-6
Q₂ = -2x10^-6
r₁³ = 3³=27
r₁=(-3,0,0)
r₂³ = 4³=64
r₂=(0,0,4)

resultant direction = r₁+r₂=(-3,0,4)
magnitude = (666, 0 , 281)
E-field at origin = (-3*666, 0 , 4*281) N/C

Also, why does r₁ suddenly become negative? In the question you're told it's (3,0,0)m, so why does it become (-3,0,0) ?

 

[bmarson123]

Sorry, I get i) now. I've recalculated it and I've got (863.2, 0, -1151.0) N. Is that a more reasonable answer?

 

[quietrain]

Sorry, I get i) now. I've recalculated it and I've got (863.2, 0, -1151.0) N. Is that a more reasonable answer?

for i)

F = kQQ/r³ * (3,0,-4)
= 9x10⁹ * (4x10^-12) / 125 * (3,0,-4)
=0.000288 * (3,0,-4) N

i don't think your answer is right since it is quite huge.

 

[quietrain]

Also, why does r₁ suddenly become negative? In the question you're told it's (3,0,0)m, so why does it become (-3,0,0) ?

because r1 is not really the r1 given in the question per say

in the question, r1 is the coordinate, that's why its (3,0,0) , pointing from origin to that coordinate.

in ii), it wants the vector electric field at the origin.

so you know that electric field points outwards from +ve charges

so if your charge were at (3,0,0) , then where would it point at the origin? (you should draw it out to understand better) anyway,

0 <----------------- 3 (x-axis)

as per the diagram above, the E-field is pointing to the left. hence, your vector is (-3,0,0)