- #1

freshcoast

- 185

- 1

A thin rod of length 2L has charge Q uniformly distributed along its length.

1. Draw a graph of the charge density as a function of x from -L to L

2. Find the electric potential at point P a distance y away

3. Use the result in part 2 of the electric field at point P (you are finding the electric field as a function of a y equation)

4. Assess ; Draw the V versus y graph and draw the E versus y graph.

2. Known equations

v = k ∫ dq / r

e = -∇v

λ = Q / L

3. Attempt

1.

graph charge density as a function of x from -L to L would just look like this because using the values and plugging it into the lambda equation it would produce these values.

2.

To find the electric potential I first need to find the correct form of dq.

Using the lambda equation

λ = Q / 2L ,

since L = 2L ,

Q = λ2L , then for dq = λ2dl

then

V = k ∫ λ2dl / √(L/2)^2 + y^2 (bounds for integral is from 0 to 2L)

because r = √(L/2)^2 + y^2

then it integrates to

[tex]V = 2kλ [ln(y^2 + √(L/2)^2 + y^2] |2L to 0 [/tex]

[tex]V = 2kλ [ln(y^2 + √((2L/2)^2 + y^2 - ln(y^2 + √(0/2)^2 + y^2))][/tex]

[tex]V = 2kλ [ln(y^2 + √L^2 + y^2 - ln(y^2 + √y^2] [/tex]

then using log rules,

[tex] V = 2kλ [ln (y^2 + √L^2 + y^2 ) / (y^2 + y) [/tex]

is this so far correct? I need to make sure this is correct before I can continue