Posted by Amanda E. on Saturday, February 19, 2011 at 10:10pm.

Two positive charges of magnitude q are each a distance d from the origin A of a coordinate system as shown above.
( Figure: h t t p : / / draw.to/DS0Qid)

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 answers to both questions are A but I don't know how to get these. Please help. Thank You.

No one has answered this question yet.

I cannot find your figure nor the "following points" that you are supposed to choose among.

h t t p : / / draw.to/DS0Qid

you need to delete the space between "h t t p : / / " to open the link since I could not post a link here... By the way, the points are drawn on the figure.. so please take a look.. and thank you!

I was able to display your figure this time. i don't know why my browser would not go there before.

At point A, the E-fields due to the two +q charges are equal and opposite, resulting in zero field. You cannot get lower than that. So A is the answer.

The E field is a minimum where the electric potential gradient (which equals the field) is zero. That would also be point A. That is also the point where the sum of q/r for the two charges is a maximum.

To determine where the electric field is least in magnitude, we can use Coulomb's Law to calculate the magnitude of the electric field at each point. Coulomb's Law states that the electric field at a point due to a positive point charge is proportionate to the charge and inversely proportionate to the distance from the charge.

To calculate the magnitude of the electric field at a point, we can use the formula:

E = k * |q| / r^2

where E is the magnitude of the electric field, k is Coulomb's constant (approximately 9 x 10^9 Nm^2/C^2), |q| is the magnitude of the charge, and r is the distance from the charge.

In this case, we have two positive charges of magnitude q that are each a distance d from the origin. So, we need to calculate the electric field at three different points - P, Q, and R, as shown in the figure.

Let's consider point P first. The electric field at point P due to the first charge is:

E1 = k * |q| / (d^2)

Similarly, the electric field at point P due to the second charge is also:

E2 = k * |q| / (d^2)

Since these two electric fields are in the same direction, their magnitudes add up:

E_P = E1 + E2 = 2 * (k * |q| / (d^2))

Similarly, we can calculate the electric field at points Q and R. For point Q, the distance from each charge is 2d, so the electric field is:

E_Q = 2 * (k * |q| / (4d^2))

For point R, the distance from each charge is sqrt(2)d, so the electric field is:

E_R = 2 * (k * |q| / (2d^2))

From these calculations, we can see that:

E_P > E_Q > E_R

Therefore, the electric field is least in magnitude at point R.

Now let's determine where the electric potential is greatest in magnitude. The electric potential at a point due to a positive point charge is proportionate to the charge and inversely proportionate to the distance from the charge. The electric potential at a point in a system of multiple charges is given by the sum of the electric potentials due to each individual charge.

To calculate the electric potential at each point, we can use the formula:

V = k * |q| / r

where V is the electric potential, k is Coulomb's constant, |q| is the magnitude of the charge, and r is the distance from the charge.

Using this formula, we can calculate the electric potential at points P, Q, and R due to each individual charge, and then sum them up to get the total electric potential at each point.

Since the electric potential is a scalar quantity, we only need to consider the magnitudes of the electric potentials at each point.

Let's consider point P first. The electric potential at point P due to the first charge is:

V1 = k * |q| / d

Similarly, the electric potential at point P due to the second charge is also:

V2 = k * |q| / d

The total electric potential at point P is:

V_P = V1 + V2 = 2 * (k * |q| / d)

Similarly, we can calculate the electric potential at points Q and R. For point Q, the distance from each charge is 2d, so the electric potential is:

V_Q = 2 * (k * |q| / (2d))

For point R, the distance from each charge is sqrt(2)d, so the electric potential is:

V_R = 2 * (k * |q| / (sqrt(2)d))

If we simplify these equations, we find:

V_P > V_Q > V_R

Therefore, the electric potential is greatest in magnitude at point P.

In summary, the electric field is least in magnitude at point R, and the electric potential is greatest in magnitude at point P.