Electron and Uniformly Charged Disk

[thebert010]

Homework Statement

An electron (e) is to be released from rest on the central axis of a uniformly charged disk of radius R. The surface charge density on the disk is +3.50 µC/m2.
(a) What is the magnitude of the electron's initial acceleration if it is released at a distance R from the center of the disk?
(b) What is the magnitude if it is released at a distance R/100 from the center?
(c) What is the magnitude if it is released at a distance R/1000 from the center?

Homework Equations

$$E(z,R)=\frac{\sigma}{2\epsilon_{0}}\left(1-\frac{z}{\sqrt{z^{2}+R^{2}}}\right)$$

The Attempt at a Solution

So I understand that the acceleration will increase slightly as I get closer to the disk. What I cannot figure out is exactly how R and Fractions of R affect the magnitude of the Electric field. I tried to assume that z=R which allowed me to say that for:
a) I showed that the R term turns into (1-(1/sqrt(2)))
Beyond that, I cannot figure out how to go about b) and c) and my online prof just keeps telling me to look at examples in the book...

Please help!

Thank You!

 

[pgardn]

Homework Statement

An electron (e) is to be released from rest on the central axis of a uniformly charged disk of radius R. The surface charge density on the disk is +3.50 µC/m2.
(a) What is the magnitude of the electron's initial acceleration if it is released at a distance R from the center of the disk?
(b) What is the magnitude if it is released at a distance R/100 from the center?
(c) What is the magnitude if it is released at a distance R/1000 from the center?

Homework Equations

$$E(z,R)=\frac{\sigma}{2\epsilon_{0}}\left(1-\frac{z}{\sqrt{z^{2}+R^{2}}}\right)$$

The Attempt at a Solution

So I understand that the acceleration will increase slightly as I get closer to the disk. What I cannot figure out is exactly how R and Fractions of R affect the magnitude of the Electric field. I tried to assume that z=R which allowed me to say that for:
a) I showed that the R term turns into (1-(1/sqrt(2)))
Beyond that, I cannot figure out how to go about b) and c) and my online prof just keeps telling me to look at examples in the book...

Please help!

Thank You!

Well it asks for acceleration. So firstly you know Fe= E*Q What other Force equation helps to find the acceleration of an electron (that has mass) so you can make an equality and find a, acceleration?

Secondly R relates the radius of the disc to how far away the electron is (Z) from the center of the charged disc. So if you made them both = 1 m, would it then be easier to calculate (or "see") the strength of the electric field at a distance Z, that is 1/100 th the radius of disc with a radius of 1 meter? or 100 x closer than the radius of the disc?

 

[thebert010]

pgardn,
You are a Godsend

Thank You So Much!

 

[pgardn]

pgardn,
You are a Godsend

Thank You So Much!

Please inform my wife of this revelation.

 

[rwisz]

Dear student,

First of all, let's clarify a few things. The problem states that the electron is released from rest at a distance R from the center of the disk. This means that the initial velocity of the electron is zero. Also, the problem does not specify the mass of the electron, so we will assume it to be the standard value of 9.11 x 10^-31 kg.

Now, let's look at the equations we have at our disposal. The electric field at a distance z from the center of a uniformly charged disk with surface charge density σ is given by:

E(z,R)= σ/2ε0 (1-z/√(z^2+R^2))

This equation tells us that the magnitude of the electric field decreases as we move away from the center of the disk (z increases). It also tells us that the electric field is directly proportional to the surface charge density and inversely proportional to the distance from the center of the disk.

To find the acceleration of the electron at a distance R from the center of the disk, we can use Newton's second law, F=ma, where F is the electric force acting on the electron. The electric force is given by:

F = qE = (-e)E

where q is the charge of the electron and e is the elementary charge, 1.60 x 10^-19 C.

So, at a distance R from the center of the disk, the acceleration of the electron is given by:

a = F/m = (-e)E/m = (-eσ/2ε0m)(1-z/√(z^2+R^2))

Substituting the given values, we get:

a = (-1.60 x 10^-19 C)(3.50 x 10^-6 C/m^2)/(2)(8.85 x 10^-12 C^2/(N·m^2))(1-R/√(R^2+R^2))

Simplifying, we get:

a = (-8.80 x 10^19)/(9.90 x 10^-12)(1-R/√2R) = -8.89 x 10^7 m/s^2 (1-R/√2R)

So, the magnitude of the electron's initial acceleration at a distance R from the center of the disk is 8.89 x 10^7