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Potential energy is the energy stored in an object due to its position within a force field, such as a gravitational or electric field. In the context of electrostatics, potential energy is related to the position of charges within an electric field. Here, when a proton is in an electric field, it possesses electric potential energy due to its charge and position between the plates.

• Electric potential energy (\( \Delta PE \)) is a form of energy that is determined by the charge and the electric potential difference (\( V \)).
• Changes in potential energy in an electrostatic system can lead to changes in kinetic energy.

When the proton is accelerated from rest, its decrease in potential energy exactly converts into kinetic energy. This is described by the energy conservation equation: \[\Delta PE = KE\]This equation shows that the difference in potential energy becomes the kinetic energy of the proton, indicating a transformation from stored energy to energy of motion.

Kinetic energy refers to the energy an object possesses due to its motion. For a proton moving in space, its kinetic energy determines how fast it's moving. The formula to calculate kinetic energy (\( KE \)) for a proton is:\[KE = \frac{1}{2} m v^2\]where \( m \) is the mass of the proton and \( v \) is its velocity.

Basically, as a proton accelerates and gains speed, its kinetic energy increases. Let's break it down:

• The mass (\( m \)) of a proton is a known constant, approximately \( 1.67 \times 10^{-27} \mathrm{~kg} \).
• The velocity (\( v \)) in this scenario is given as \( 10^6 \mathrm{~m/s} \).
• Substituting these values, we calculate its kinetic energy; here, it comes out to be \( 8.35 \times 10^{-14} \mathrm{~J} \).

This kinetic energy is crucial in determining how much potential energy is necessary to accelerate a stationary proton to its target speed due to the potential difference between plates.

Proton acceleration in this context involves moving a hydrogen ion (proton) from rest to a specified speed by applying a potential difference. The speed mentioned here, \( 10^6 \mathrm{~m/s} \), requires careful calculations of energy transitions.

The steps to understand proton acceleration involve:

• Starting with the proton at rest implies its initial kinetic energy is zero.
• As the proton is subjected to an electric field, it experiences a force due to its charge.
• This force causes the proton to accelerate, increasing its speed and hence its kinetic energy.

The acceleration happens because the electric field applies a force on the proton, converting electrical potential energy into kinetic energy. At the new speed, the proton's kinetic energy has been calculated using the mass and velocity relevant to the proton, allowing us to determine the required potential difference (\( V \)). To create the speed \( 10^6 \mathrm{~m/s} \), after solving with the formula derived from energy conservation, the potential difference is estimated to be close to \( 5000 \mathrm{~V} \). This means the electric field does the work needed to achieve this level of acceleration.