Question 23.2: The Electric Field of a Uniform Ring of Charge A ring of rad...
The Electric Field of a Uniform Ring of Charge
A ring of radius a carries a uniformly distributed positive total charge Q. Calculate the electric field due to the ring at a point P lying a distance x from its center along the central axis perpendicular to the plane of the ring (Fig. 23.3a).
Conceptualize Figure 23.3a shows the electric field contribution d \overrightarrow{E} at P due to a single segment of charge at the top of the ring. This field vector can be resolved into components d E_x parallel to the axis of the ring and d E_{\perp} perpendicular to the axis. Figure 23.3b shows the electric field contributions from two segments on opposite sides of the ring. Because of the symmetry of the situation, the perpendicular components d E_{\perp} of the field cancel. That is true for all pairs of segments around the ring, so we can ignore the perpendicular component of the field and focus solely on the parallel components d E_x, which simply add.
Categorize Because the ring is continuous, we are evaluating the field due to a continuous charge distribution rather than a group of individual charges.
Analyze Evaluate the parallel component of an electric field contribution from a segment of charge dq on the ring:
(1) d E_x=k_e \frac{d q}{r^2} \cos \theta=k_e \frac{d q}{a^2+x^2} \cos \theta
From the geometry in Figure 23.3a, evaluate cos θ:
(2) \cos \theta=\frac{x}{r}=\frac{x}{\left(a^2+x^2\right)^{1 / 2}}
Substitute Equation (2) into Equation (1):
d E_x=k_e \frac{d q}{a^2+x^2}\left\lfloor\frac{x}{\left(a^2+x^2\right)^{1 / 2}}\right\rfloor=\frac{k_e x}{\left(a^2+x^2\right)^{3 / 2}} d qAll segments of the ring make the same contribution to the field at P because they are all equidistant from this point. Integrate over the circumference of the ring to obtain the total field at P:
E_x=\int \frac{k_e x}{\left(a^2+x^2\right)^{3 / 2}} d q=\frac{k_e x}{\left(a^2+x^2\right)^{3 / 2}} \int d q(3) E=\frac{k_e x}{\left(a^2+x^2\right)^{3 / 2}} Q
Finalize The electric field at P is of this magnitude and directed along the x axis, away from the ring. This result shows that the field is zero at x = 0. Is that consistent with the symmetry in the problem? Furthermore, notice that Equation (3) reduces to k_e Q / x^2 if x >> a, so the ring acts like a point charge for locations far away from the ring. From a faraway point, we cannot distinguish the ring shape of the charge.