Problem 2 Electric field and force A positively charged wire is bent into a semicircle of radius R,as shown in the figure below. 0 R The total charge on the semicircle is O.However,the charge per unit length along the semicircle is non-uniform and given by入=，cos8. a) What is the relationship between R and O? b)If a particle with a charge g is placed at the origin,what is the total force on the particle? Show all your work including setting up and integrating any necessary integrals. Answer. (a)In order to find a relation between R and it is necessary to integrate the charge density A because the charge distribution is non-uniform Q=∫h=名，eos0Ra6=队sm9rn=2队. wire (b)The force on the charged particle at the center P of the semicircle is given by F(P)=qE(P). The electric field at the center P of the semicircle is given by (P)= ATEo wire The unit vector,f,located at the field point,is directed from the source to the field point and in Cartesian coordinates is given by f=-sin@'i-cos'j. 1010 Problem 2 Electric field and force A positively charged wire is bent into a semicircle of radius R , as shown in the figure below. The total charge on the semicircle is Q . However, the charge per unit length along the semicircle is non-uniform and given by 0 ! = ! cos" . a) What is the relationship between !0 , R and Q ? b) If a particle with a charge q is placed at the origin, what is the total force on the particle? Show all your work including setting up and integrating any necessary integrals. Answer. (a) In order to find a relation between !0 , R and Q it is necessary to integrate the charge density ! because the charge distribution is non-uniform / 2 / 2 0 0 / 2 0 / 2 cos sin 2 wire Q ds Rd R R ! " ! " ! " ! " # # ! ! # ! # $= $= $=% $=% = = $ $ = $ = & & . (b) The force on the charged particle at the center P of the semicircle is given by F(P) = qE(P) ! ! . The electric field at the center P of the semicircle is given by 2 0 1 ( ) ˆ 4 wire ds P r ! "# = $ E r ! . The unit vector, rˆ , located at the field point, is directed from the source to the field point and in Cartesian coordinates is given by ˆ ˆ rˆ = #sin!" i # cos!" j