Two positive charges of magnitude q are each a distance d from the origin A of a coordinate system as shown above.
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At which of the following points is the electric field least in magnitude?
At which of the following points is the electric potential greatest in magnitude?
The answersto both questions are A but I don't know how to get these. Please help. Thank You.
The electric field at point A is least in magnitude because the two charges have the same magnitude and are equidistant from point A, so the electric field vectors from each charge will cancel each other out.
The electric potential at point A is greatest in magnitude because the electric potential is the sum of the potentials from each charge, and since the charges have the same magnitude, the potential at point A will be twice the potential from either charge.
To determine the point where the electric field is least in magnitude, we can use the principle that the electric field from a point charge decreases with increasing distance.
Since the charges are positive, the electric field is directed radially outward from each charge. Thus, we can compare the magnitudes of the electric fields produced by each charge at the different points.
To find the point where the electric field is least, we want to minimize the sum of the magnitudes of the electric fields from each charge.
Let's consider each given point one by one:
Point A: At this point, the electric fields from both charges have equal magnitudes and opposite directions. The sum of the magnitudes of the electric fields at point A is 2E, where E is the magnitude of the electric field produced by each charge.
Point B: The distance between point B and each charge is greater than at point A. Therefore, the magnitudes of the electric fields from both charges at point B will be smaller than at point A. Thus, the sum of the magnitudes of the electric fields at point B will be less than 2E.
Point C: Similar to at point B, the distances between point C and each charge are greater than at point A. Therefore, the magnitudes of the electric fields from both charges at point C will be smaller than at point A, resulting in a sum less than 2E.
Point D: The distance between point D and one of the charges is greater than at point A, but the other charge is at the same distance. Therefore, one of the electric fields will be smaller, but the other will remain the same. Thus, the sum of the magnitudes of the electric fields at point D will be between 2E and 3E.
Therefore, based on our analysis, the point where the electric field is least in magnitude is Point A.
Now, let's find the point where the electric potential is greatest in magnitude. Note that the electric potential depends on both the magnitude of the charges and the distance from the charges.
Since the charges are positive, the electric potential will be positive as well. The electric potential decreases with increasing distance from a positive charge.
Considering each given point:
Point A: Both charges are at a distance d from point A. Therefore, the contribution to the electric potential from each charge will be the same. As the point is closer to both charges, the electric potential will be greater compared to the other points.
Point B: The distance from one of the charges is greater than at point A, resulting in a smaller contribution to the electric potential. Therefore, the electric potential at point B will be less than at point A.
Point C: Similar to at point B, the distances from both charges are greater than at point A, resulting in smaller contributions to the electric potential. Thus, the electric potential at point C will be less than at point A.
Point D: The distance from one of the charges is greater than at point A, resulting in a smaller contribution to the electric potential. Therefore, the electric potential at point D will be less than at point A.
Based on our analysis, the point where the electric potential is greatest in magnitude is Point A.
Thus, the answers to both questions are A.
To determine where the electric field is least in magnitude, we need to consider the concept of superposition. The electric field at a point due to multiple charges is the vector sum of the electric fields produced by each individual charge.
In this case, we have two positive charges of magnitude q located at distances d from the origin. Let's label these charges as Q1 and Q2. For simplicity, let's assume that both charges have the same magnitude q.
To find where the electric field is least in magnitude, we need to consider the direction and magnitude of the electric fields produced by each charge at different points in the coordinate system.
At point A, which is the origin of the coordinate system, the electric fields produced by Q1 and Q2 have the same magnitude but opposite directions. Therefore, they will cancel out each other, resulting in a net electric field of zero. So, the electric field at point A is least in magnitude.
To determine where the electric potential is greatest in magnitude, we can use the formula for electric potential. The electric potential at a point due to a single charge is given by the equation:
V = k(Q/r)
Where V is the electric potential, k is the Coulomb constant, Q is the charge, and r is the distance from the charge.
Since we have two charges of magnitude q, we can calculate the electric potential at different points by taking into account the superposition principle.
At point A, which is the origin, the electric potential due to Q1 and Q2 is zero since the net charge at that point is zero (as discussed earlier). Therefore, the electric potential at point A is zero, which means it is the greatest in magnitude among the given points.
In summary, the electric field is least in magnitude at point A, and the electric potential is greatest in magnitude at point A.