Final answer:
The phenomenon of a zero electric field one-fourth of the way between two charges indicates that the magnitudes of the charges are different and that they have opposite signs. If q1 is designated as the closer charge, it must be smaller in magnitude and opposite in sign to q2.
Step-by-step explanation:
Understanding Electric Fields Between Two Charges
When considering two point charges (q1) and (q2), and the electric field one-fourth of the way from q1 to q2 is zero, one can analyze the scenario using Coulomb's law and the properties of electric fields. If the electric field is zero at a point between the two charges, this suggests that the electric fields due to each charge are equal in magnitude but opposite in direction at that point, effectively canceling each other out. Given that the point in question is closer to q1 than to q2, it implies that q1 is smaller in magnitude than q2. Furthermore, for the electric fields from both charges to cancel each other out, the charges must have opposite signs.
To illustrate, suppose q1 is closer to the point where the electric field is zero, and let q2 be farther away. Since electric field strength decreases with the square of the distance, for the field to be zero one-fourth of the way from q1, q1 must have a smaller magnitude due to it having less distance to cancel out q2's larger field. If both charges had the same sign, the fields would add up rather than cancel out. Therefore, without loss of generality, if q1 is positive, q2 would need to be negative, and vice versa, in order to achieve the zero electric field at that point. Option a correctly reflects that q1 is positive and q2 is negative, assuming we designate q1 as the closer charge.