Physics Challenge for Boris Korsunsky, Column Editor ... · 07/03/2019 · U CV= and the charge on...
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Physics Challenge for Teachers and Students Boris Korsunsky, Column Editor Weston High School, Weston, MA 02493 [email protected] Solution to March 2013 Challenge w Washing the Plates Two parallel vertical metal plates, shaped as squares with side h, are held above the surface of a nonconducting liquid of density r so that their bottom edges touch the surface of the liquid. The plates are a distance d apart. After the plates are connected to a battery that maintains a constant voltage V, the liquid rises between the plates, barely reaching their top edges. Neglecting the effects of surface tension, find the dielectric constant k of the liquid. Solution: When the capacitor is connected to the bat- tery, the energy stored in the air-filled capacitor is 2 1 , 2 U CV = and the charge on each plate is q = CV. When the capacitor is filled with the dielectric liquid, its capacitance becomes kC, where k is the dielectric constant of the liquid. This increases the charge stored on each plate to kCV. The additional charge of (k – 1) CV came through the battery, which did some work, given by ΔqV = (k – 1)CV 2 . However, the electrical energy stored in the capacitor only increases by ½ kCV 2 – ½ CV 2 = ½ (k – 1)CV 2 . The other half of the work done by the battery goes into raising the liquid into the capacitor. An amount of liquid of volume h 2 d and mass rh 2 d is raised into the capacitor. The center of mass of the liq- uid is raised to the midpoint of the capacitor, a distance of h/2. So the increase in the gravitational potential energy of the liquid is mgΔy = rh 3 dg/2. The work done by the battery is equal to the change in energy of the system, namely the increase in the capacitor’s stored electrical energy and the increase in the gravitational P.E. of the liquid. 2 2 3 1 1 ( 1) ( 1) . 2 2 k CV k CV gdh r − = − + A little algebra gives 3 2 1 . gdh k CV r = + But C = ε 0 h 2 /d, where ε 0 is the vacuum permittivity, so 2 2 0 1 . gd h k V r ε = + (Contributed by Grant Volle, Booker T. Washington HS, Tulsa, OK) Column Editor’s note: Some readers understood that the water reached the top of the plates at the instant of its maximum rise (the intended interpretation) and some thought it was the equilibrium elevation. We regret the imperfect wording of the problem and, since it reasonably allowed for both interpretations, we give credit to the solu- tions of both types. Some of the successful solvers are: Phil Cahill (The SI Organization, Inc., Rosemont, PA) Don Easton (Lacombe, Alberta, Canada) Fernando Ferreira (Universidade da Beira Interior, Covilhã, Portugal) Art Hovey (Milford, CT) José Ignacio Íñiguez de la Torre (Universidad de Salaman- ca, Salamanca, Spain) José Costa Leme (High School Lanheses, Viana do Castelo, Portugal) Carl E. Mungan (U. S. Naval Academy, Annapolis, MD) Thomas Olsen (Society of Physics Students, AIP, College Park, MD) Jason L. Smith (Richland Community College, Decatur, IL) Cássio dos Santos Sousa, student (Instituto Tecnológico de Aeronáutica, São Paulo, Brazil) Many thanks to all contributors and we hope to hear from many more of you in the future! Guidelines for contributors: – We ask that all solutions, preferably in Word format, be submitted to the dedicated email address [email protected]. Each message will receive an au- tomatic acknowledgment. – The subject line of each message should be the same as the name of the solution file (see the instructions be- low). – The deadline for submitting the solutions is the last day of the corresponding month. – We can no longer guarantee that we’ll publish every successful solver’s name; each month, a representative selection of names will be published, both in print and on the web. – If your name is—for instance—Joseph Ratzinger, please name the file “May13Ratzinger” (do not include THE PHYSICS TEACHER ◆ Vol. 51, 2013
Transcript of Physics Challenge for Boris Korsunsky, Column Editor ... · 07/03/2019 · U CV= and the charge on...
Physics Challenge for Teachers and Students
Boris Korsunsky, Column EditorWeston High School, Weston, MA 02493 [email protected]
Solution to March 2013 Challenge
w Washing the Plates
Two parallel vertical metal plates, shaped as squares with side h, are held above the surface of a nonconducting liquid of density r so that their bottom edges touch the surface of the liquid. The plates are a distance d apart. After the plates are connected to a battery that maintains a constant voltage V, the liquid rises between the plates, barely reaching their top edges. Neglecting the effects of surface tension, find the dielectric constant k of the liquid.
Solution: When the capacitor is connected to the bat-tery, the energy stored in the air-filled capacitor is
21 ,2
U CV=
and the charge on each plate is q = CV.
When the capacitor is filled with the dielectric liquid, its capacitance becomes kC, where k is the dielectric constant of the liquid. This increases the charge stored on each plate to kCV. The additional charge of (k – 1)CV came through the battery, which did some work, given by ΔqV = (k – 1)CV2. However, the electrical energy stored in the capacitor only increases by ½ kCV2 – ½ CV2 = ½ (k – 1)CV2. The other half of the work done by the battery goes into raising the liquid into the capacitor.An amount of liquid of volume h2d and mass rh2d is raised into the capacitor. The center of mass of the liq-uid is raised to the midpoint of the capacitor, a distance of h/2. So the increase in the gravitational potential energy of the liquid is mgΔy = rh3dg/2. The work done by the battery is equal to the change in energy of the system, namely the increase in the capacitor’s stored electrical energy and the increase in the gravitational P.E. of the liquid. 2 2 31 1( 1) ( 1) .
2 2k CV k CV gdhr− = − +
A little algebra gives3
21 .gdhkCVr
= +
But C = ε0h2/d, where ε0 is the vacuum permittivity, so 2
20
1 .gd hkV
rε
= +
(Contributed by Grant Volle, Booker T. Washington HS, Tulsa, OK)
Column Editor’s note: Some readers understood that the water reached the top of the plates at the instant of its maximum rise (the intended interpretation) and some thought it was the equilibrium elevation. We regret the imperfect wording of the problem and, since it reasonably allowed for both interpretations, we give credit to the solu-tions of both types. Some of the successful solvers are: Phil Cahill (The SI Organization, Inc., Rosemont, PA)Don Easton (Lacombe, Alberta, Canada)Fernando Ferreira (Universidade da Beira Interior,
Covilhã, Portugal)Art Hovey (Milford, CT)José Ignacio Íñiguez de la Torre (Universidad de Salaman-
ca, Salamanca, Spain)José Costa Leme (High School Lanheses, Viana do Castelo,
Portugal)Carl E. Mungan (U. S. Naval Academy, Annapolis,
MD) Thomas Olsen (Society of Physics Students, AIP, College
Park, MD)Jason L. Smith (Richland Community College, Decatur,
IL) Cássio dos Santos Sousa, student (Instituto Tecnológico de
Aeronáutica, São Paulo, Brazil)Many thanks to all contributors and we hope to hear from many more of you in the future!
Guidelines for contributors:
– We ask that all solutions, preferably in Word format, be submitted to the dedicated email address [email protected]. Each message will receive an au-tomatic acknowledgment.
– The subject line of each message should be the same as the name of the solution file (see the instructions be-low).
– The deadline for submitting the solutions is the last day of the corresponding month.
– We can no longer guarantee that we’ll publish every successful solver’s name; each month, a representative selection of names will be published, both in print and on the web.
– If your name is—for instance—Joseph Ratzinger, please name the file “May13Ratzinger” (do not include
The Physics Teacher ◆ Vol. 51, 2013
The Physics Teacher ◆ Vol. 51, 2013
your first initial) when submitting the May solution. – If you have a message for the Column Editor, you
may contact him at [email protected]; however, please do not send your solutions to this address.
As always, we look forward to your contributions and hope that they will include not only solutions but also your own Challenges that you wish to sub-mit for the column.
Boris Korsunsky, Column Editor
