Example 10  The Electric Fields from Separate Charges May Cancel

Two positive point charges, q1=+16 mC and q2=+4.0 mC, are separated in a vacuum by a distance of 3.0 m, as Figure 18.20 illustrates. Find the spot on the line between the charges where the net electric field is zero.

The two point charges q1 and q2 create electric fields E1 and E2 that cancel at a location P on the line between the charges.
Figure 18.20  The two point charges q1 and q2 create electric fields E1 and E2 that cancel at a location P on the line between the charges.

Reasoning  Between the charges the two field contributions have opposite directions, and the net electric field is zero at the place where the magnitude of E1 equals that of E2. However, since q2 is smaller than q1, this location must be closer to q2, in order that the field of the smaller charge can balance the field of the larger charge. In the drawing, the cancellation spot is labeled P, and its distance from q1 is d.

Problem solving insight
Equation 18.3 gives only the magnitude of the electric field produced by a point charge. Therefore, do not use negative numbers for the symbol |q| in this equation.

Solution At P, E1=E2, and using the expression E=k|q|/r2, we have

Rearranging this expression shows that 4.0(3.0 md)2=d2. Taking the square root of each side of this equation reveals that
The plus and minus signs on the right occur because either the positive or negative root can be taken. Therefore, there are two possible values for d: +2.0 m and +6.0 m. The value +6.0 m corresponds to a location off to the right of both charges, where the magnitudes of E1 and E2 are equal, but where the directions are the same. Thus, E1 and E2 do not cancel at this spot. The other value for d corresponds to the location shown in the drawing and is the zero-field location: .



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