PHY6 9-8-6 Exam Solution y 4 3 6 x. A central particle of charge 3 i urrounded by a hexagonal array of other charged particle (>). The length of a ide i, and charge are placed at each corner. (a) [6 point] Find the component of the force along the x-axi (F x ) on the central particle. The force along the horiontal direction and along the direction from bottom left to upper right cancel. Thu, only the force from the upper left charge and bottom right charge matter. The upper left charge i greatet in magnitude, and the force i attractive ince the middle charge i negative. Thu: ( 4)( 3) ( )( 3) Fx = K co 6 K co 6 6 Fx = K =3K (b) [6 point] Find the component of the force along the y-axi (F y ) on the central particle. ( 4)( 3) ( )( 3) Fx = K in 6 K in 6 6 3 Fx = K = 3 3K Page of 9
PHY6 9-8-6 y λ λ R x. Conider electric charge ditributed along a one-dimenional path in the form hown a two ection of ¼ of a circle each. The circle i centered at the origin with a radiu of R, and the linear charge denity i λ in the left uadrant and λ in the right. (a) [6 point] Find the component of the electric field along the x-axi (E x ) at the origin (,). The field contribution from the left arc eual that of the right arc. So we can calculate twice the x-component of the field from the right arc: d π / λrdθ Ex = K coθ = K co R θ R λ π / Kλ Ex = K inθ = R R (b) [6 point] Find the component of the electric field along the y-axi (E y ) at the origin (,). The field cancel in the y direction. Page of 9
PHY6 9-8-6 y x 3. Conider a cube with ide length = m and one corner at the origin (,,) a hown. (a) [6 point] What i the total charge encloed by the cube if the electric field i E= 3x ˆ i 5 ˆ j k ˆ N/C? ( ) ( ) ( ) ε 3 3 3 ( ) ( x ) enc 6 Φ= = Φ i= i =Φ = Φ = ε = ε 3 = 3 =.8 C enc = = 3 other eual becaue field in thoe direction i con (b) [6 point] What i the electric charge denity (C/m 3 ) at the center of the right face at x = m if the electric field i the ame a in part (a)? ρ ε ε E E x y E = E = = x x y ε = ε = = By the way, we could integrate thi to get the anwer for part (a) enc 3 enc ε ( 3 ) 3 ( 6x ) ( 6 ) 8.85 C/m ( 6 ) ( 3 ) = ρdv = ε dy d dx x = ε = Page 3 of 9
PHY6 9-8-6 R 4. [8 point] A flat nonconducting urface infinite in extent carrie a uniform charge 9 denity of σ = 3 C/m. A mall circular hole of radiu R =.5 m ha been cut in the middle of the heet a hown. Calculate the electric field at a point = 5 m away from the center of the hole along an axi perpendicular to the urface. (In other word, conider R, but don t et R / exactly eual to ero. You may find the uperpoition principle ueful.) Uing the uperpoition principle, add the field due to an infinite heet of charge (without a hole) plu a mall dik of negative charge (ame magnitude but oppoite ign charge denity). E= Eheet Ehole σ Eheet = ˆ ε E ˆ hole = K when viewed far away R = R πr σ σ R E ˆ hole = = 4πε ε σ R σ E= ˆ =.955 ˆ = 6 ˆ N/C ε ε ( ) and where ( π )( σ ) Page 4 of 9
PHY6 9-8-6 5. [8 point] An electron i launched away from the urface of an infinite 6 nonconducting heet of charge with a velocity of v = m/on a trajectory perpendicular to the urface. The charge denity of the heet i σ = 5 nc/m. I the electron able to reach a ditance infinitely far away from the charged heet, and if not, how far doe it travel before turning around? The charge of the electron i 9 3 = e=.6 C, and the electron ma i = 9. kg. Becaue the electron i negatively charged and the heet poitive, the electron i attracted to the heet by a force σ F= E, where E = x ˆ taking x in the direction of the particle (left). ε Thu, it will top and turn around. The difference in electric potential from the urface of the heet to poition x i: σ Δ V = x ε σ Δ U = Δ V = ( e) x ε At the turn-around v m e Δ U = K initial eσ x= mv ε ε mv eσ = =.4 m Page 5 of 9
PHY6 9-8-6 6. A conducting phere of radiu R contain a charge Q. It i urrounded by a concentric pherical conducting hell of radiu R > R and charge Q. (a) [6 point] What i the difference in electric potential between the hell and the phere? -Q Q E Q E = K rˆ R < r < R r by Gau' L:aw and ymmetry Δ V = E d E d = E d= E dr R Q Q Δ V = K dr = K = KQ R R r r R R R (b) [6 point] What i the capacitance of the arrangement of conductor? Q= CΔV Q= ΔV K R R RR C = K R R ( ) Page 6 of 9
PHY6 9-8-6 7. [6 point] Initially two electron are fixed in place with a eparation of.5 µm. How much work mut be done to bring a third electron in from infinity to complete an euilateral triangle? Note that for a point charge, the electric potential a ditance r away i V = K r The work to place a negative charged particle a ditance r away from point charge alo of negative charge i the invere of the change in the potential energy: e W =Δ U =Δ V = K = r.4 J 8. [6 point] Sketch the electric field line for a negatively charged particle above the urface of a flat perfectly conducting urface (both above and below the urface). conductor Page 7 of 9
PHY6 9-8-6 9. A.5 µf capacitor i charged to a potential difference of V, and the charging battery i diconnected. (a) [6 point] What i the energy tored in the capacitor? = CΔ V = 6 (.5 F)( ) 4 U = = C ( Δ V) =.8 J C (b) [6 point] If the charged capacitor i then connected in parallel with a econd (initially uncharged) capacitor, and if the potential difference acro the firt capacitor ubeuently drop to 9 V, what i the capacitance of thi econd capacitor? Charge cannot be detroyed, o it remain on the repective plate even though they are connected to additional plate. But the capacitance mut change (capacitor in parallel), o the potential difference mut change to maintain contant : C = C C e (parallel) C Δ V = = = Δ Ce C C C C rearranging: ΔV C = C =.5μF ΔV V Page 8 of 9
PHY6 9-8-6. [6 point] The electric potential along the x-axi (in V) i plotted veru the value of x, (in cm). Evaluate the x-component of the electrical force (in Newton, including ign) on a proton located on the x-axi at x = cm. The electric field i given by the negative rate of change of the electric potential per unit length: ΔV 5 V E = = =5 V/m (from graph) Δx. m F = E = (.6 9 C)( 5 V/m) = xˆ 6 o F 4. N Force i directed in negative x direction.. [6 point] Two wire are made out of the ame material (copper). One ha a circular cro ection with radiu r = mm and a length of cm, the other ha a uare cro ection with width = mm and a length of 5 cm. Which wire ha the larger reitance? L R = ρ A L ρ. ρ 5 circular: R = ρ = = π 6 r π π L ρ.5 ρ ρ uare: R = ρ = = > 6 π π Suare wire ha larger reitance. 5 5 Page 9 of 9