Figure 22-38a shows two charged particles fixed in place on an x-axis with separation L. The ratio of their charge magnitudes is . Figure 22-38b shows the x component of their net electric field along the x-axis just to the right of particle 2. The x-axis scale is set by . (a) At what value of is maximum? (b) If particle 2 has charge , what is the value of that maximum?
Using the concept of the electric field at a given point, we can get the value of an individual electric field by a charge. Again for the maximum value of the net field, we can differentiate the electric field equation for getting the value of x. Now, substituting the value of x, we can get the value of the required electric field.
The magnitude of the electric field, (1)
where R = The distance of field point from the charge, and q = charge of the particle
According to the superposition principle, the electric field at a point due to more than one charge,
For it to be possible for the net field to vanish at some x > 0, the two individual fields (caused by and ) must point in opposite directions for x > 0. They are therefore oppositely charged considering their positions. Further, since the net field points more strongly leftward for the small positive x (where it is very close to ), then we conclude that localid="1657282744540" is the negative-valued charge. Thus, is a positive-valued charge.
From the given ratio, we can now considered for getting a maximum electric field that
Thus using equation (1), we can find the individual fields, and substituting this into equation (2), we can get the net electric as:
Setting at (see graph) x = 20 cm , the graph immediately leads to To get the maximum value of the electric field, we can differentiate the above equation for x, and equating it to zero, we can get the value of x as:
Hence, the value of x is 34 cm
Substituting the given values in equation (3), we can the maximum electric field for the charge of particle 2, as follows:
Hence, the value of the electric field is
In Fig. 22-56, a “semi-infinite” non-conducting rod (that is, infinite in one direction only) has uniform linear charge density l. Show that the electric field at point P makes an angle of with the rod and that this result is independent of the distance R. (Hint: Separately find the component of parallel to the rod and the component perpendicular to the rod.)
A thin non-conducting rod with a uniform distribution of positive charge Q is bent into a complete circle of radius R (Fig. 22-48). The central perpendicular axis through the ring is a z axis, with the origin at the center of the ring. What is the magnitude of the electric field due to the rod at (a) and (b) ? (c) In terms of R, at what positive value of z is that magnitude maximum? (d) If and , what is the maximum magnitude?
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