Questions: Which set has the lowest potential energy? A) a +2 charge and a +2 charge separated by distance of 2x B) a +2 charge and a +2 charge separated by distance of x C) a +2 charge and a -2 charge separated by distance of 2x D) a +2 charge and a -2 charge separated by distance of x

Solution Steps

Electric potential energy between two point charges is given by the formula:

\[ U = k \frac{q_1 q_2}{r} \]

where \( U \) is the potential energy, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. The potential energy is positive if both charges have the same sign and negative if they have opposite signs.

Let's calculate the potential energy for each option:

• Option A: Two +2 charges separated by distance \( 2x \). \[ U_A = k \frac{(+2)(+2)}{2x} = k \frac{4}{2x} = \frac{2k}{x} \]

• Option B: Two +2 charges separated by distance \( x \). \[ U_B = k \frac{(+2)(+2)}{x} = k \frac{4}{x} \]

• Option C: A +2 charge and a -2 charge separated by distance \( 2x \). \[ U_C = k \frac{(+2)(-2)}{2x} = k \frac{-4}{2x} = \frac{-2k}{x} \]

• Option D: A +2 charge and a -2 charge separated by distance \( x \). \[ U_D = k \frac{(+2)(-2)}{x} = k \frac{-4}{x} \]

Now, compare the potential energies calculated:

• \( U_A = \frac{2k}{x} \)
• \( U_B = \frac{4k}{x} \)
• \( U_C = \frac{-2k}{x} \)
• \( U_D = \frac{-4k}{x} \)

The lowest potential energy is the most negative value. Among these, \( U_D = \frac{-4k}{x} \) is the most negative.

Final Answer