What are the magnitude and direction of the electric field that will stop these protons in a distance of 1.40 m
The Correct Answer and Explanation is:
Magnitude: 1.45 x 10⁴ N/C
Direction: Opposite to the direction of the proton’s velocity
Explanation
To find the magnitude and direction of the electric field, we can apply the work-energy theorem. This theorem states that the net work done on an object is equal to the change in its kinetic energy (W = ΔKE).
First, let’s determine the work required to stop the protons. The change in kinetic energy is the final kinetic energy (KE_final) minus the initial kinetic energy (KE_initial). Since the protons are brought to a stop, their final kinetic energy is zero.
ΔKE = KE_final – KE_initial = 0 – (3.25 × 10⁻¹⁵ J) = -3.25 × 10⁻¹⁵ J
The work done (W) by the electric field must be equal to this change in kinetic energy, so W = -3.25 × 10⁻¹⁵ J.
The work done by a constant electric force (F) over a distance (d) is given by the formula W = F * d * cos(θ). To stop the protons, the electric force must act in the direction opposite to their motion. Therefore, the angle θ between the force and the displacement is 180°, and cos(180°) = -1. This gives W = -F * d.
Now we can relate the force to the kinetic energy:
-3.25 × 10⁻¹⁵ J = -F * (1.40 m)
F = (3.25 × 10⁻¹⁵ J) / (1.40 m) = 2.3214 × 10⁻¹⁵ N
The electric force on a charged particle is related to the electric field (E) by the equation F = qE, where q is the charge of the particle. For a proton, the charge is q = 1.602 × 10⁻¹⁹ C. We can rearrange this to solve for the magnitude of the electric field:
E = F / q
E = (2.3214 × 10⁻¹⁵ N) / (1.602 × 10⁻¹⁹ C)
E ≈ 14490 N/C or 1.449 × 10⁴ N/C
Rounding to three significant figures, the magnitude of the electric field is 1.45 × 10⁴ N/C.
For the direction, a proton has a positive charge. The electric force on a positive charge is in the same direction as the electric field. As established, the force must oppose the proton’s velocity to stop it. Therefore, the electric field must also be directed opposite to the direction of the proton’s velocity.
