Charged Particles in Electric Fields: Energy, Potential, and Work

Electric Field Fundamentals

5.a) Charged Particle Movement and Energy Changes

When a charged particle moves in an electric field, if the electric force (F_e) is conservative, the change in mechanical energy (ΔEM) is zero. This implies an inverse relationship between kinetic energy (Ec) and potential energy (Ep): if Ec increases, Ep decreases, and vice versa.

If a particle's speed decreases upon entering an electric field, its charge is negative. Conversely, if its speed increases, its charge is positive.

5.b) Work Done by Magnetic Fields on Charges

No work is performed by a magnetic field on a charged particle. A magnetic field generates a magnetic force that is always perpendicular to the path of the charged particle, meaning it does no work.

6.a) Potential Energy Change for a Negative Charge

For a negatively charged particle passing through a point with potential V_A, its electric potential energy (E_P) is given by E_P = Vq. Since the electric force (F_e) is conservative, the change in mechanical energy (ΔEM) is zero. This implies that if kinetic energy (Ec) increases, potential energy (Ep) decreases. Therefore, a negatively charged particle moving freely under the action of the electric field will always decrease its electric potential energy.

6.b) Work Done on Equipotential Surfaces

No work is done to transfer a charge (positive or negative) from point C to point D if both points lie on the same equipotential surface. On an equipotential surface, all points have the same electric potential, so the potential difference between C and D is zero. Consequently, the work done (W_C→D) is zero.

15.a) Potential and Field Intensity Near Negative Charge

True. When moving away from a negative charge, the electric potential increases (becomes less negative). This is because the potential due to a negative charge is negative, and as distance increases, its absolute value decreases, making it closer to zero (and thus higher). However, the electric field intensity, which is inversely proportional to the square of the distance (1/r²), always decreases as the distance from the charge increases.

15.b) Vanishing Potential vs. Non-Zero Electric Field

False. If the electric potential (V) vanishes at a point, it means the sum of potentials from all charges at that point is zero (V = ΣV_i = 0). For instance, if two charges q₁ and q₂ are equal (q₁ = q₂), and their individual potentials at a point are equal (V₁ = V₂), then the total potential V would be 2V₁. For V to vanish, V₁ would have to be zero, which typically occurs at infinite distance where the field intensity also vanishes. Therefore, the statement that potential vanishes but field intensity does not, is not universally true and can be false in certain contexts.

16.a) Negative Charge Motion and Potential Energy

The electric field intensity points in the direction of decreasing electric potential. When a negative charge moves freely under the action of the electric field, it moves in the direction opposite to the electric field. This means it moves from a region of higher electric potential to a region of lower electric potential. Consequently, its electric potential energy (E_P = qV) will decrease, as V decreases and q is negative (making qV more negative).

16.b) Positive Charge Motion and Potential Energy

If the charge were positive, its behavior would differ. A positive charge moving freely under the action of the electric field moves in the same direction as the electric field. This means it moves from a region of higher electric potential to a region of lower electric potential. As its velocity increases, its kinetic energy (Ec) increases. Since the electric force is conservative (ΔEM = 0), its electric potential energy (Ep) must decrease. This also implies a decrease in electric potential (V), as Ep = qV and q is positive.

19.a) Positive Charge Dynamics in Electric Fields

The electric field is a conservative field, meaning the change in mechanical energy (ΔEM) is zero.

• If a positive charge moves in the same direction as the electric field, its velocity increases, leading to an increase in kinetic energy (Ec). Consequently, its electric potential energy (Ep) decreases.
• If it moves in the opposite direction, its velocity will decrease, resulting in a decrease in kinetic energy (Ec). In this case, its electric potential energy (Ep) will increase.

19.b) Work in Closed Loops in Electric Fields

If a charged particle describes a closed circular path within an electric field, the total work done by the electric field is zero. This is because the electric field is a conservative force field, meaning the work done is independent of the path taken and depends only on the initial and final points. For a closed loop, the initial and final points are the same, so the change in potential energy (ΔPE) is zero, and thus the work done is zero. The change in potential energy does not depend on the direction of the field, but on the potential difference between the points.