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When an alpha particle, a type of charged particle, enters a magnetic field, its motion is affected by the field. Imagine pushing something into a stream of water; the flow will change its path. Similarly, a magnetic field exerts a force on charged particles, causing them to move in a circular path. This is because the magnetic field applies a force that is always perpendicular to the particle's velocity, pulling it into a curved motion.

An alpha particle, with a charge of +2e, can be significantly influenced by a magnetic field. The strength of the effect depends on several factors:

• The magnitude of the magnetic field, which in our example is 2.20 Tesla (T).
• The charge on the particle, which is the fundamental charge multiplied by 2 for alpha particles.
• The velocity of the particle as it enters the field.

In problems involving a magnetic field, it's crucial to remember that these factors work together to determine the particle's behavior and how much it will curve as it moves through the field.

The potential difference, also known as voltage, is fundamental in propelling an alpha particle forward. When the particle is accelerated from rest, it gains energy from the electric field caused by the potential difference. Think of it like a hill that the particle rolls down, gaining speed as it goes.

In mathematical terms, the potential energy provided by this electric field is equal to the charge times the potential difference (qV). For an alpha particle, this equation looks like:

• The charge \( q = +2e \)
• The potential difference \( V = 1.20 \times 10^6 \, \text{V} \)

This potential energy is then converted entirely into kinetic energy as the particle starts moving. This concept explains why the alpha particle has a specific speed when entering the magnetic field, directly related to the voltage; higher the voltage, higher the speed.

Once the alpha particle is moving through the magnetic field, it travels in a circular path. This is due to the centripetal force. Imagine spinning a ball on a string; the string pulls the ball towards the center as it moves in a circle. Similarly, the magnetic force acts as the 'string', providing the centripetal force that keeps the particle moving in its curved path.

Mathematically, centripetal force is given by the expression:

• \( F_c = \frac{mv^2}{r} \)

Where:

• \( m \) is the mass of the particle
• \( v \) is its velocity
• \( r \) is the radius of the circle

In the presence of a magnetic field, this centripetal force is exactly equal to the magnetic force acting on the particle, which is calculated as \( qvB \). Understanding this balance helps us determine the radius of the path; knowing how these forces work together is key to predicting the motion of charged particles in magnetic fields.

Kinetic energy is what an alpha particle gains when it is accelerated by a potential difference. This energy is the energy of motion, which means the faster the particle moves, the more kinetic energy it has.

The relationship between kinetic energy and speed can be defined using this equation:

• \( K.E. = \frac{1}{2}mv^2 \)

Where:

• \( m \) is the mass of the alpha particle
• \( v \) is its speed

The entire energy imparted to the particle through the voltage (potential difference) is converted into kinetic energy. Because energy is conserved, the initial potential energy equals the final kinetic energy, allowing us to solve for the particle's speed. This conversion outlines how the alpha particle reaches its maximum speed before entering the magnetic field, showing us the direct effect of the potential difference on kinetic energy.