What are the magnitude and direction of the electric field that will stop these protons in a distance of 1.40 m? Magnitude Direction —Select— N/C
The Correct Answer and Explanation is:
To determine the magnitude and direction of the electric field that will stop the protons in a distance of 1.40 meters, we need to use the concepts of kinetic energy, electric potential energy, and Newton’s second law of motion.
Step 1: Kinetic Energy of the Proton
When the proton is moving, it has kinetic energy. The kinetic energy (KEKEKE) of the proton is given by the formula: KE=12mv2KE = \frac{1}{2}mv^2KE=21mv2
where:
- mmm is the mass of the proton (1.67×10−27 kg1.67 \times 10^{-27}\, \text{kg}1.67×10−27kg)
- vvv is the velocity of the proton.
If the proton is to be stopped by the electric field, the electric field must do work equal to the initial kinetic energy of the proton.
Step 2: Work Done by Electric Field
The work (WWW) done by the electric field on the proton is given by: W=F⋅dW = F \cdot dW=F⋅d
where:
- FFF is the force exerted by the electric field.
- ddd is the distance over which the force acts (1.40 m1.40 \, \text{m}1.40m).
The force exerted by the electric field is: F=qEF = qEF=qE
where:
- qqq is the charge of the proton (1.6×10−19 C1.6 \times 10^{-19} \, \text{C}1.6×10−19C)
- EEE is the electric field strength (the quantity we need to find).
The work done by the electric field is equal to the change in the kinetic energy, so: KE=F⋅d=qE⋅dKE = F \cdot d = qE \cdot dKE=F⋅d=qE⋅d
Substitute the expression for KEKEKE: 12mv2=qE⋅d\frac{1}{2}mv^2 = qE \cdot d21mv2=qE⋅d
Solve for the electric field EEE: E=mv22qdE = \frac{mv^2}{2qd}E=2qdmv2
Step 3: Magnitude of the Electric Field
To calculate the exact magnitude of EEE, we would need the initial velocity of the proton (vvv). Without that value, we can’t proceed with the exact calculation. However, if you provide the proton’s velocity, we can substitute it into this formula to find the electric field.
Direction of the Electric Field
The direction of the electric field will be opposite to the motion of the proton because the field is doing negative work to stop the proton. Since protons are positively charged, the field must point in the direction opposite to the proton’s motion in order to decelerate it.
