Uniformly charged ring on the axis

[SlyCedix]

Homework Statement

The electric field on the axis of a uniformly charged ring has magnitude 360 kN/C at a point 6.2 cm from the ring center. The magnitude 19 cm from the center is 150 kN/C ; in both cases the field points away from the ring.

Homework Equations

1. Find the ring's radius.
2. Find the ring's charge.

The Attempt at a Solution

E=kxQ/(x² + a²)^3/2
360000 = kQ(.062)/(.062²+a²)^3/2
150000 = kQ(.19)/(.19²+a²)^3/2

360/150 = (.062/.19) * ((.062²+a²)/(.19²+a²))^3/2

(360/150 * .19/.062)^2/3 = (.062²+a²)/(.19²+a²)

(360/150 * .19/.062)^2/3 * (.19²+a²)= .062²+a²

(360/150 * .19/.062)^2/3 * (.19²) - .062² = a² - (360/150 * .19/.062)^2/3 * a²)

-0.00247871714 = -2.7819469745 a^2

a = sqrt(0.00247871714/2.7819469) = .030 m = 3.0 cm

Which turned out to be incorrect, and with an incorrect radius, I've got nothing to go off of for charge

 

[SammyS]

Homework Statement

The electric field on the axis of a uniformly charged ring has magnitude 360 kN/C at a point 6.2 cm from the ring center. The magnitude 19 cm from the center is 150 kN/C ; in both cases the field points away from the ring.

Homework Equations

1. Find the ring's radius.
2. Find the ring's charge.

The Attempt at a Solution

E=kxQ/(x² + a²)^3/2
360000 = kQ(.062)/(.062²+a²)^3/2
150000 = kQ(.19)/(.019²+a²)^3/2

360/150 = (.062/.19) * ((.062²+a²)/(.019²+a²))^3/2

(360/150 * .19/.062)^2/3 = (.062²+a²)/(.019²+a²)

(360/150 * .19/.062)^2/3 * (.019²+a²)= .062²+a²

(360/150 * .19/.062)^2/3 * (.019²) - .062² = a² - (360/150 * .19/.062)^2/3 * a²)

-0.00247871714 = -2.7819469745 a^2

a = sqrt(0.00247871714/2.7819469) = .030 m = 3.0 cm

Which turned out to be incorrect, and with an incorrect radius, I've got nothing to go off of for charge

Hello @SlyCedix . Welcome to PF.It's just a little Algebra mistake.

What do you get when you divide ##\displaystyle \ \frac{kQ (0.062)} {(0.062^2+a^2)^{3/2}} \ ## by ##\displaystyle \ \frac{kQ (0.19)} {(0.19^2+a^2)^{3/2}} \ ##?

 

[SlyCedix]

Wow, I cannot believe I missed that.

So the revised eq would be:

$$\frac{360}{150} = \frac{.062}{.19} * (\frac{(.19^2+a^2)}{(.062^2+a^2)})^\frac{3}{2}$$

$$(\frac{360}{150} * .\frac{.062}{.19})^\frac{2}{3} = \frac{(.19^2+a^2)}{(.062^2+a^2)}$$

$$(\frac{360}{150} * \frac{.062}{.19})^\frac{2}{3}* (.062^2+a^2)= .19^2+a^2$$

$$(\frac{360}{150} * \frac{.062}{.19})^\frac{2}{3} * .062^2 - .19^2 = a^2 - (\frac{360}{150} * \frac{.062}{.19})^\frac{2}{3} a^2$$

$$0.033212 = -2.7819469745 a^2$$

~~But that would mean a is non real, despite there being a real answer.~~

Scratch that, put it in my calculator wrong, it's actually

$$-0.02156 = -2.7819469745 a^2$$

Which is still wrong, but closer

 

[SammyS]

Wow, I cannot believe I missed that.

So the revised eq would be:

$$\frac{360}{150} = \frac{.062}{.19} * (\frac{(.19^2+a^2)}{(.062^2+a^2)})^\frac{3}{2}$$

$$(\frac{360}{150} * .\frac{.062}{.19})^\frac{2}{3} = \frac{(.19^2+a^2)}{(.062^2+a^2)}$$

$$(\frac{360}{150} * \frac{.062}{.19})^\frac{2}{3}* (.062^2+a^2)= .19^2+a^2$$

$$(\frac{360}{150} * \frac{.062}{.19})^\frac{2}{3} * .062^2 - .19^2 = a^2 - (\frac{360}{150} * \frac{.062}{.19})^\frac{2}{3} a^2$$

$$0.033212 = -2.7819469745 a^2$$

But that would mean a is non real, despite there being a real answer.

You made a similar mistake as previously, when going from the first to the second line .

 

[SlyCedix]

That was it, thanks, answer was 8.8cm

 

[SammyS]

That was it, thanks, answer was 8.8cm

Good.

What total charge did you get ?

 

[SlyCedix]

800 nC