A total charge Q = 0.1 mu C is distributed uniformly over a quarter circle arc o

Question

A total charge Q = 0.1 mu C is distributed uniformly over a quarter circle arc of radius a = 7.2 cm as shown. What is lambda the linear charge density along the arc? What is E_x, the value of the x-component of the electric field at the origin (x, y) = (0, 0)? What is E_y, the value of the y component of the electric field at the origin (x, y) = (0, 0)? How does the magnitude of the electric field at the origin for the quarter circle arc you have just calculated compare to the electric field at the origin produced by a point charge Q = 0.1 mu C located a distance a = 7.2 cm from the origin along a 45 degree line as shown in the figure? The magnitude of the field from the point charge is less than that from the quarter arc of charge. The magnitude of the field from the point charge is equal to that from the quarter arc of charge The magnitude of the field from the point charge is greater than that from the quarter arc of charge What is the magnitude of the electric field at the origin produced by a semi-circular arc of charge = 0.2 mu C, twice the charge of the quarter circle arc?

Explanation / Answer

a) Linear charge density is equal to charge over the length.
So,
Lambda = Q / (1/4*2*pi*a)
=> Lambda = 0.1e-6/(0.25*2*pi*7.2e-2)
=> Lambda = 8.84e-7 C/m

b)Ex = k*Lambda / a
=> Ex = (8.99*10^9*8.84e-7)/7.2e-2
=> Ex = 11.04e4 N/C

c) This is the same formula from b as it is a circle and should have uniform x and y components because it is a circle
Ey = 11.04e4 N/C

d) Option c

e) Since we already know Ey for the quarter arc, the Ey for the semicircle will be twice that. The Ex for the two parts of the semicircle cancel each other out.
So, E = 2*Ey = 22.08e4 N/C

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