{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Fo nt 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 255 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "PHYSICS RESOURCE PACKETS P ROJECT File: DIELCAP.MWS" }}{PARA 0 "" 0 "" {TEXT -1 110 "Rose-Hulman Institute of Technology \+ Authors: Perry Peters & Greg Williby" }}{PARA 0 "" 0 "" {TEXT -1 97 "http://www.rose-hulman.edu/~moloney \+ Software: Maple V Release 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 515 "The electric field insid e concentric conducting spheres of radius R1 and R2 (R2>R1) is reduced by a factor of 2 when a particular dielectric material fills the enti re space between R1 and R2. This means that the potential difference \+ between the spheres is also reduced by a factor of 2, and the capacita nce is increased by a factor of 2. (The charge Q on the spheres is kep t the same.) When the dielectric doesn't fill the entire space between the spheres, the capacitance is increased by less than a factor of 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "a) \+ How thick would a dielectric spherical shell of thickness d, extending from R1 to R1+d have to be in order to increase the capacitance by a \+ factor 1.5? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "b) Would this thickness be the same if it extended from R2-d to R2?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "So lution" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Part A: Determine kappa" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Determine k by using the \+ relationship the E=Eo/kappa." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq1 :=E=Eo/kappa;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Since the field \+ is reduced by a factor of 2, the following is true:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eq2:=subs(E=Eo/2,eq1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Now solve for kappa." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "kappa=solve(eq2,kappa);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Pa rt B: Determine the dielectric's thickness" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Electric field between the sphe res:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Efield:=q/(4*Pi*epsilon[0]* r^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "where V is the differen ce in potential between the two conducting spheres. V can be expresse d in terms of a constant, Q, R1 and R2 by the following formula:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "V:=Int(Efield,r=R1..R2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "V:=int(Efield,r=R1..R2);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The capacitance without the diele ctric is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "C:=simplify(q/V);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "When a dielectric is added, the po tential difference becomes:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Delt a*'V'=Delta*'Vo'/kappa;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "Find \+ the potential between the two spheres again, but this time divide the \+ potential by kappa for r=R1..R1+d, and then add on the regular potenti al from r=R1+d..R2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Vdie:=int(Ef ield,r=R1..R1+d)/kappa+int(Efield,r=R1+d..R2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The capacitance with the dielectric is:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "Cdie:=simplify(q/Vdie);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 109 "Now solve for the thickness of the dielectric (d) that causes the capacitance to increase by a factor of 1.5." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol:=d=simplify(solve(Cdie=3/2*C,d));" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Substitute in the value of kappa \+ obtained in Part A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sub:=subs(ka ppa=2,sol);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Perform a sanity c heck. If R1 was 1 and R2 was 2, then d should be between 0 and 1." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(\{R1=1,R2=2\},sub);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Part C" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Now find out what the result would be if the dielectric extended from R2-d to R2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Use the regular potential for r=R1..R2-d, and then divide by kappa for r=R2-d..R2." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 59 "Vdie2:=int(Efield,r=R1..R2-d)+int(Efield,r=R 2-d..R2)/kappa;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The capacitanc e with the dielectric on the outer shell is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Cdie2:=simplify(q/Vdie2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Now solve for the thickness of the dielectric (d) that c auses the capacitance to increase by a factor of 1.5." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol2:=d=simplify(solve(Cdie2=3/2*C,d));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Substitute in the value of kappa o btained in Part A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sub2:=subs(ka ppa=2,sol2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Perform a sanity \+ check. If R1 was 1 and R2 was 2, then d should be between 0 and 1." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\{R1=1,R2=2\},sub2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The thickness of the dielectric is dependent on where it is placed between the two spheres." }}}}{MARK " 19 0 0" 3 }{VIEWOPTS 1 1 0 1 1 1803 }