Magnitude of electric field at a point problem

[roam]

Homework Statement

If Q = 80 nC, a = 3.0 m, and b = 4.0 m in the figure, what is the magnitude of the electric field at point P?

[PLAIN]http://img19.imageshack.us/img19/1596/picat.jpg

Homework Equations

$$\vec{E} = k_e \frac{|q|}{r^2}$$

$$\left\{\begin{matrix}\vec{E}_x=k_e \frac{|q|}{a^2+b^2} \cos \ \phi \\ \vec{E}_y=k_e \frac{|q|}{a^2+b^2} \sin \ \phi\end{matrix}\right.$$

k_e = 8.9 x 10⁹ (Coulomb's constant)

The Attempt at a Solution

The charge -Q only has an x component, the electric field due to this charge is:

$$\vec{E}_{(-Q),x}= (8.9 \times 10^9) \times \frac{-80 \times 10^{-9}}{4^2}=-44.5$$

The two "2Q" charges cancel out each other's y-components, so we need to only consider their x components:

$$\vec{E}_{(2Q), x} = (8.9 \times 10^{9}) \times \frac{2 \times (80 \times 10^{-9})}{3^2+4^2} \cos 45 = 40.27$$

I got the 45 degree angle is from here:

[PLAIN]http://img200.imageshack.us/img200/5499/pic2ae.jpg

So the total electric field is:

$$\vec{E}_{(-Q), x}+2\vec{E}_{(2Q), x}=-44.5+ 2\times(40.27)=36.05 N/C$$

But this is wrong since the correct answer must be 47 N/C. What's wrong with my calculation? Any help is greatly appreciated.

 

[ehild]

why 45°?

ehild

 

[cupid.callin]

check angle again ...

tan 45 = 1

is a=b? no

do it again

 

[roam]

Thank you very much guys, I got it.

 

[rwisz]

Your calculation is incorrect because you have not taken into account the distance between the charges and point P. The correct formula for the electric field at point P is:

\vec{E} = k_e \frac{|q|}{r^2} \hat{r}

Where r is the distance between the charge and point P, and \hat{r} is the unit vector in the direction from the charge to point P. In this case, you need to calculate the distance between each charge and point P separately, and then calculate the electric field due to each charge using the formula above. Finally, you can add the electric fields due to each charge using vector addition to get the total electric field at point P.

In addition, your calculation for the electric field due to the -Q charge is incorrect. The correct formula is:

\vec{E}_{(-Q)} = k_e \frac{|q|}{r^2} \hat{r}

Where r is the distance between the -Q charge and point P, and \hat{r} is the unit vector in the direction from the -Q charge to point P. In this case, the distance r is not equal to 4m, as you have assumed. You need to use the Pythagorean theorem to calculate the distance r.