# If the velocity of the charged particle is doubled and the value of the magnetic field is halved, then the radius will be

The radius of a charged particle moving in a magnetic field is given by the equation:

r = mv / (qB)

Where r is the radius of the particle's circular path, m is the mass of the particle, v is its velocity, q is its charge, and B is the magnetic field strength.

If the velocity of the charged particle is doubled (v → 2v) and the value of the magnetic field is halved (B → B/2), then the radius of the particle's circular path will be:

r' = m(2v) / (q(B/2))

Simplifying the expression, we get:

r' = mv / (qB/2)

r' = 2(mv / (qB))

r' = 2r

Therefore, the radius will be doubled (r → 2r)

Other methods

Full explanation

Sure! Here's a full explanation of why the radius of a charged particle moving in a magnetic field will double when the velocity is doubled and the magnetic field is halved.

When a charged particle moves through a magnetic field, it experiences a force known as the magnetic force. This force is perpendicular to both the particle's velocity and the magnetic field, and it causes the particle to move in a circular path.

The radius of this circular path is determined by several factors, including the mass of the particle (m), the charge of the particle (q), the velocity of the particle (v), and the strength of the magnetic field (B).

The equation for the radius of the circular path is:

r = mv / (qB)

Now, let's consider what happens when the velocity of the charged particle is doubled and the magnetic field is halved. We can represent these changes with the following substitutions:

v → 2v

B → B/2

Using the original equation for the radius, we can substitute in these new values:

r' = m(2v) / (q(B/2))

Now we can simplify the equation:

r' = 2mv / (qB)

We can see that the only change from the original equation is the coefficient in front of the mv term. This coefficient is now 2, which means that the new radius is twice as large as the original radius.

Therefore, we can conclude that when the velocity of a charged particle is doubled and the magnetic field is halved, the radius of the particle's circular path will double as well.