Consider a charged ring of radius 27 cm and total charge 15 nC.

We are interested in the electric field a perpendicular distance z away from the center of the ring.

At what distance from the center of the ring does the electric field become maximum?

Hint: The field for a ring of charge is:

Ering = kQz/ (z^2+R^2)^3/2

I understand that we would need to differentiate this equation and solve for when it equals zero..but I'm having real hard time differentiating. Could someone help me out?

This is what I have so far...

E'ring = kQ (z/[(z^2 +R^2)^3/2])
= {{1/(z^2+R^2)^3/2}} - {{z(z^2+R^2)^1/2}} / {{(z^2+R^2)^3}}

This is what I got by using the product rule. But I don't know how to get rid of the constant R from the equations and or how to simplify in such a way that I would be able to solve for Z.

Firstly, you are to use not the product rule, but the quotient rule, check out the related wikipedia article. Secondly, there is no necessity to "get rid" of R. Your solution can depend on R, for example. Just treat R as a constant (In fact it is, Radius (R) = 2.7 times 10^(-1) m)

You made a number of errors.

= {{1/(z^2+R^2)^3/2}} - {{2z^2}} / {{(z^2+R^2)^5/2}}

set it equal to zero. multiply both sides by (z^2+R^2)^3/2

0=1-2z/(z^2+R^2)
so 2z=z^2+R^2

z^2-2z+R^2=0

z=(2+-sqrt(4-4R))/2=1+-sqrt(1-R)

check my math.

To find the distance from the center of the ring where the electric field becomes maximum, you need to differentiate the equation for the electric field with respect to z and solve for when it equals zero. Let's go through the differentiation step by step.

The equation for the electric field of a charged ring is:

Ering = kQz / (z^2 + R^2)^(3/2)

where:
k is the electrostatic constant
Q is the total charge of the ring
z is the perpendicular distance from the center of the ring
R is the radius of the ring

Now, let's differentiate the equation with respect to z:

E'ring = d/dz (kQz / (z^2 + R^2)^(3/2))

To differentiate this, we can use the quotient rule. The quotient rule states that if we have two functions u(z) and v(z):

d/du (u/v) = (v*du - u*dv) / v^2

Let's apply the quotient rule to differentiate the equation:

E'ring = [(z^2 + R^2)^(3/2) * d/dz (kQz) - kQz * d/dz (z^2 + R^2)^(3/2)] / (z^2 + R^2)^3

Now, let's differentiate each term separately:

1. d/dz (kQz)
Since kQ is a constant, its derivative with respect to z is zero. So, d/dz (kQz) = 0.

2. d/dz (z^2 + R^2)^(3/2)
To differentiate this, we can use the chain rule. Let u = z^2 + R^2, then the function becomes u^(3/2). The derivative of u^(3/2) with respect to u is (3/2)u^(1/2). But since u = z^2 + R^2, we multiply by the derivative of u with respect to z, which is 2z:

d/dz (z^2 + R^2)^(3/2) = (3/2)*(z^2 + R^2)^(1/2) * 2z = 3z*(z^2 + R^2)^(1/2)

Now let's substitute these derivatives back into the equation:

E'ring = [(z^2 + R^2)^(3/2) * 0 - kQz * 3z*(z^2 + R^2)^(1/2)] / (z^2 + R^2)^3

Simplifying this equation, we get:

E'ring = -3kQz^2 / (z^2 + R^2)^(5/2)

Now, to find the distance from the center of the ring where the electric field becomes maximum, we set E'ring = 0 and solve for z:

-3kQz^2 / (z^2 + R^2)^(5/2) = 0

Since the numerator is zero, we can ignore it and solve for when the denominator equals zero:

(z^2 + R^2)^(5/2) = 0

This equation does not have any real solutions because the square of any real number (z^2 + R^2) is always positive. Hence, there is no maximum electric field along the axis of the ring.

Note: Differentiation can be a confusing process, but by taking it step by step and using the appropriate rules, we can calculate the derivative of a given function.