Results 1 to 2 of 2

Math Help - Non conducting disc?

  1. #1
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539

    Non conducting disc?

    A non-conducting disc of radius a and uniform positive charge density \sigma is placed on the ground, with its axis vertical. A particle of mass m and positive charge q is dropped, along the axis of the disc, from a height H with zero initial velocity. The particle has \frac{q}{m} = \frac{4\epsilon_0 g}{\sigma}

    A) Find the value of H if the particle just reaches the disc.
    Answer:
    Spoiler:
    H = \frac{4a}{3}


    B) Sketch the potential energy of the particle as a function of its height and find its equilibrium position.

    How to do it?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Aug 2009
    Posts
    143
    Voltage at the particle:
    V_P=\int^{a}_{0}\frac{\sigma 2\pi rdr}{4\pi\epsilon_0\sqrt{r^2+H^2}}=\frac{\sigma}{2 \epsilon_0}(\sqrt{a^2+H^2}-H) (1)

    Voltage at the disc center [simply set H=0 in (1)]:
    V_O=\frac{\sigma}{2\epsilon_0}{a} (2)

    Increase in electric potential:
    \Delta U=q(V_O-V_P)=\frac{q\sigma}{2\epsilon_0}(a+H-\sqrt{a^2+H^2}) (3)

    Decrease in gravitational potential:
    \Delta V=mgH (4)

    Energy conservation:
    \Delta U=\Delta V (5)

    (3)(4)(5) together give
    \frac{q\sigma}{2\epsilon_0}(a+H-\sqrt{a^2+H^2})=mgH (6)

    Use \frac{q}{m}=\frac{4\epsilon_0 g}{\sigma} leads to
    H+a-\sqrt{a^2+H^2}=\frac{H}{2}

    Therefore H=\frac{4a}{3}

    Luobo
    Last edited by mr fantastic; September 19th 2009 at 01:56 AM. Reason: Restored original reply
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Conducting Math Research
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: August 11th 2011, 05:52 AM
  2. Replies: 0
    Last Post: February 8th 2011, 11:54 AM
  3. Conducting a Better Statistics Study
    Posted in the Statistics Forum
    Replies: 2
    Last Post: January 20th 2010, 01:56 PM
  4. tiny spheres conducting
    Posted in the Advanced Applied Math Forum
    Replies: 9
    Last Post: February 26th 2008, 08:21 PM
  5. If the area of a disc...
    Posted in the Geometry Forum
    Replies: 3
    Last Post: October 27th 2006, 08:52 AM

Search Tags


/mathhelpforum @mathhelpforum