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Chapter 20: Problem 8

The ring in Example 20.6 carries total charge \(Q,\) and the point \(P\) is the same distance \(r=\sqrt{x^{2}+a^{2}}\) from all parts of the ring. So why isn't the electric field of the ring just \(k Q / r^{2} ?\)

Short Answer

Expert verified
The electric field of the ring is not just \(kQ/r^2\) because the electric field is a vector quantity, and the different field vectors from each point of the ring don't all point in the same direction. The perpendicular components of the vectors cancel out and the field along the axis of the symmetrically charged ring is \(kQx/(x^2 + a^2)^(3/2)\).

Step by step solution

01

Understand the Electric Field Due to a Ring of Charge

The result \(kQ/r^2\) would have applied if all the individual field vectors due to the ring of charge were pointing in the same direction, such is the case for a point charge. However, for a ring of charge, each individual electric field vector points in a different direction (proceeding radially outward from each 'piece' of the ring). These vectors need to be summed up!
02

Evaluate the Sum of Individual Field Vectors

The vectors from charges diametrically across the ring from each other cancel out in the horizontal (perpendicular) direction because they have equal magnitudes but opposite directions. This means only the vertical (along the axis of the ring) components add up while the horizontal ones cancel out.
03

Final Consideration

Considering the above, we see that the actual electric field at point \(P\) isn't just \(kQ/r^2\). The true electric field due to a uniformly charged ring along its axis of symmetry is \(kQx/(x^2 + a^2)^(3/2)\), where \(k\) is Coulomb's constant, \(Q\) is the total charge on the ring and \(x\) is the distance from the center of the ring.

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Most popular questions from this chapter

A \(1.0-\mu C\) charge and a \(2.0-\mu \mathrm{C}\) charge are \(10 \mathrm{cm}\) apart. Find a point where the electric field is zero.

Three identical charges \(q\) form an equilateral triangle of side \(a\) with two charges on the \(x\) -axis and one on the positive \(y\) -axis. (a) Find an expression for the electric field at points on the \(y\) -axis above the uppermost charge. (b) Show that your result reduces to the field of a point charge \(3 q\) for \(y \gg a\).

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