Four charges are arranged at the corners of a regular tetrahedron (a pyramid with four sides that are identical equilateral triangles). The charges are 1 Coulomb each, and the sides of the tetrahedron are all 1 meter. How much energy in Joules is required to assemble this arrangement of charge
Simple. What is the Potential energy of each charge? Well, PE is a scalar, so the distances are the same to each charge, so
energy=k(q1Q2/r+Q1Q3/r+Q1Q4/r+Q2Q3/r+Q2Q4/r+Q3Q4/r)= 6k if all q are 1, all r is 1.
Now consider putting the thing together.
first charge, no energy
second charge, energy= kQq/r=k
third charge, energy= 2k
fourth charge, energy=3k
total work: 6k
To calculate the energy required to assemble the charges in the given arrangement, we can use the formula for potential energy. The potential energy between two charges is given by the equation:
U = k * q1 * q2 / r
Where U is the potential energy, k is the Coulomb constant (9 × 10^9 N⋅m²/C²), q1 and q2 are the charges, and r is the distance between the charges.
In this case, we have four charges arranged at the corners of a tetrahedron. Let's number the charges as q1, q2, q3, and q4. Since the charges are 1 Coulomb each, q1 = q2 = q3 = q4 = 1 C.
Now, we need to calculate the distance between the charges (r). In a regular tetrahedron, the distance between any two charges at the corner is equal to the length of one side of an equilateral triangle. Given that the sides of the tetrahedron are all 1 meter, the distance between any two charges is 1 meter.
Using this information, we can calculate the energy required to assemble the arrangement:
For q1 and q2: U12 = k * q1 * q2 / r = (9 × 10^9 N⋅m²/C²) * (1 C) * (1 C) / (1 m) = 9 × 10^9 J
For q1 and q3: U13 = U14 = 9 × 10^9 J (since the charges and distances are the same)
For q2 and q3: U23 = U24 = 9 × 10^9 J (since the charges and distances are the same)
For q3 and q4: U34 = 9 × 10^9 J (since the charges and distances are the same)
Now, let's calculate the total energy required to assemble the arrangement by summing up the energies between all pairs of charges:
Total energy = U12 + U13 + U14 + U23 + U24 + U34
= 9 × 10^9 J + 9 × 10^9 J + 9 × 10^9 J + 9 × 10^9 J + 9 × 10^9 J + 9 × 10^9 J
= 54 × 10^9 J
= 54 GJ (gigaJoules)
Therefore, the energy required to assemble this arrangement of charges is 54 gigajoules (GJ).