A charge of #-1 C# is at the origin. How much energy would be applied to or released from a # -1 C# charge if it is moved from # (3 ,1 ) # to #(5 ,4 ) #?

Question

A charge of #-1  C# is at the origin. How much energy would be applied to or released from a # -1 C# charge if it is moved from # (3  ,1  ) # to #(5  ,4   )  #?

Charles Aeschliman · Accepted Answer

The energy released is #=1.44*10^9J#

Potential energy is

#U=k(q_1q_2)/r#

The charge #q_1=-1C#

The charge #q_2=-1C#

The Coulomb's constant is #k=9*10^9Nm^2C^-2#

The separation

#r_1=sqrt((3)^2+(1)^2)=sqrt10m#

The separation

#r_2=sqrt((5)^2+(4)^2)=sqrt(41)#

Consequently,

#U_1=k(q_1q_2)/r_1#

#U_2=k(q_1q_2)/r_2#

#DeltaU=U_2-U_1=k(q_1q_2)/r_2-k(q_1q_2)/r_1#

#=k(q_1q_2)(1/r_2-1/r_1)#

#=9*10^9*((-1)*(-1))(1/sqrt41-1/(sqrt10))#

#=-1.44*10^9J#

The energy released is #=1.44*10^9J#

Adam Arnold · Answer

undefined The energy applied to or released from a -1 C charge when moved from (3, 1) to (5, 4) can be calculated using the formula:

$$ W = \dfrac{k \cdot q_1 \cdot q_2}{r_f - r_i} $$

Where:
- $ W $ is the work done or energy transferred,
- $ k $ is Coulomb's constant ($ 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 $),
- $ q_1 $ and $ q_2 $ are the charges ($ -1 \, \text{C} $ for both charges),
- $ r_f $ is the final position,
- $ r_i $ is the initial position.

Using the given coordinates, the distance between the initial and final positions ($ r_f - r_i $) is calculated using the distance formula:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Substitute the values into the formula to find the distance, then plug into the energy equation to find the work done.