Positive electric charge Q is distributed uniformly along a line with length 2a, lying along the y-axis between y=-a and y=+a. Find the electric field at point P on the x-axis at a distance x from the origin a dE
Positive electric charge Q is distributed uniformly along a line with length 2a, lying along the y-axis between y=-a and y=+a. Find the electric field at point P on the x-axis at a distance x from the origin. y x O x Q a -a P 2 2 r = x + y dE
Solution: de 1 do Q dQ=ndy=.dy a 4z52r24z02(x2+y2) O xa] de =de cos a 462a(x2+y y)√x+ 4丌62ax2+y
dy a Q dQ dy 2 = = ( ) 2 2 0 2 0 4 4 2 1 a x y Q dy r dQ dE + = = ( ) ( ) 3 2 2 2 0 2 2 2 2 4 0 2 4 2 cos a x y Q xdy x y x a x y Q dy dEx dE + = + + = = Solution:
de =-de sin a 4z2d(x2+y2)x2+y24752a( E 4z52aJ(2+y2 21/2 4E0 xx+a? E 0 4T60 2ad-a(x2+ E 4sx√x2+a
( ) ( ) 3 2 2 2 0 2 2 2 2 4 0 2 4 2 sin a x y Q ydy x y y a x y Q dy dEy dE + = − + + = − = − ( ) 2 2 0 3 2 2 2 0 1 4 2 4 1 x x a Q x y dy a Qx E a a x + = + = − ( ) 0 4 2 1 3 2 2 2 0 = + = − − a a y x y ydy a Q E i x x a Q E ˆ 1 4 2 2 0 + =
Discussion: When x is much larger than a(that is, x>>a), a) E i→E 4sx√x2+a 兀0 This is same as the field of a point charge b) When a is much larger than x(that is, a>>x) A=2/2a E 2IE xv(x/a)+1 2IEox c)Infinite line of charge E Field due to a power line 2丌Er
a) This is same as the field of a point charge. b) When a is much larger than x (that is, a>>x), ( ) i x i x x a E ˆ 2 ˆ 1 2 1 0 2 0 + = = Q 2a r E 0 2 = c) Infinite line of charge Discussion: When x is much larger than a (that is, x>>a), + = i x x a Q E ˆ 1 4 2 2 0 i x Q E ˆ 4 1 2 0 = Field due to a Power Line
A ring conductor with radius a carries a total charge Q uniformly distributed around it. Find the electric field at a point P that lies on the axis of the ring at a distance x from its center X+ a CdE
A ring conductor with radius a carries a total charge Q uniformly distributed around it. Find the electric field at a point P that lies on the axis of the ring at a distance x from its center. x y O a Q P x dQ 2 2 r = x + a dE