An electron is accelerated by a 4.7 kV potential difference. How strong a magnetic field must be experienced by the electron if its path is a circle of radius 8.0 cm?
Use conservation of energy to compute the velocity, v.
(1/2) m v^2 = e V
v = sqrt (2V*e/m)
V is the 4700 volt accelerating potential. e amd m are the electron's charge and mass.
If R is the radius of the circular path,
m v^2/R = e v B
Solve for B
Well, if the electron is into magnetism, I guess it's time for some "attractive" physics! Let's find out the magnetic field that the electron should experience.
To start with, we know that the electron is moving in a circle of radius 8.0 cm. The force required to keep it in circular motion is provided by the magnetic force, which is given by the equation F = B * q * v. Here, B represents the magnetic field, q is the charge of the electron (which is approximately -1.6 x 10^-19 C), and v is the velocity of the electron.
Now, the velocity of the electron can be calculated using the potential difference given. The formula for the kinetic energy of an electron is K.E. = (1/2) * m * v^2, where m is the mass of the electron (approximately 9.11 x 10^-31 kg). Since the electron is accelerated by the potential difference, we can equate the kinetic energy to the potential energy gained, which is q * V. Thus, (1/2) * m * v^2 = q * V.
Solving for v, we get v = sqrt((2 * q * V) / m).
Now that we know the velocity, we can substitute it back into the equation F = B * q * v. The magnetic force should provide the necessary centripetal force, which is m * v^2 / r, to keep the electron in circular motion.
Setting these equal, we get B * q * v = m * v^2 / r.
Simplifying this expression, we find that B = (m * v) / (q * r).
Plugging in the given values, we have B = [(9.11 x 10^-31 kg * sqrt((2 * -1.6 x 10^-19 C * 4.7 x 10^3 V) / 9.11 x 10^-31 kg))] / [-1.6 x 10^-19 C * 8.0 x 10^-2 m].
Now, I could go on with the calculations, but I think I just shocked myself with all these numbers! So, instead, let's just use a calculator and the right formula to find the magnetic field value for this electron in circular motion!
Whoa, this magnetic field question really took me for a spin. I hope my answer sparks your curiosity!
To determine the magnetic field strength experienced by the electron, we can use the formula for the centripetal force acting on a charged particle moving in a magnetic field.
The centripetal force (Fc) is given by:
Fc = (mass of the electron) × (velocity of the electron)^2 / (radius of the circular path)
Since the electron is accelerated by the potential difference, we can calculate its velocity (v) using the formula:
Potential difference (V) = (velocity of the electron) × (charge of the electron)
Rearranging this formula to solve for the velocity:
(velocity of the electron) = (Potential difference) / (charge of the electron)
Substituting the given values:
(velocity of the electron) = (4.7 kV) / (charge of the electron)
To find the magnetic field strength (B), we can rearrange the formula for the centripetal force:
Fc = (charge of the electron) × (velocity of the electron) × (magnetic field strength)
Substituting the known values:
(charge of the electron) × (velocity of the electron) × (magnetic field strength) = (mass of the electron) × (velocity of the electron)^2 / (radius of the circular path)
Simplifying:
(charge of the electron) × (magnetic field strength) = (mass of the electron) × (velocity of the electron) / (radius of the circular path)
Finally, substituting the known values into the formula:
(charge of the electron) × (magnetic field strength) = (mass of the electron) × (4.7 kV) / (charge of the electron) / (8.0 cm)
Since the charge of the electron cancels out, we are left with:
magnetic field strength = (mass of the electron) × (4.7 kV) / (8.0 cm)
Now we can calculate the magnetic field strength using the known values for the mass of the electron (9.11 × 10^-31 kg) and the conversion factor for kilovolts to volts (1 kV = 1000 V).
mass of the electron = 9.11 × 10^-31 kg
Potential difference = 4.7 kV = 4.7 × 10^3 V
radius of the circular path = 8.0 cm = 0.08 m
Substituting these values:
magnetic field strength = (9.11 × 10^-31 kg) × (4.7 × 10^3 V) / (0.08 m)
To determine the strength of the magnetic field experienced by an electron moving in a circular path, we can use the equation for the centripetal force acting on the electron, which is provided by the magnetic force. The equation is given by:
F = (mv^2) / r
Where:
F is the centripetal force,
m is the mass of the electron,
v is the velocity of the electron, and
r is the radius of the circular path.
We know that the centripetal force experienced by the electron is provided by the magnetic force, which is given by:
F = qvB
Where:
F is the magnetic force,
q is the charge of the electron,
v is the velocity of the electron, and
B is the magnetic field strength.
Since we want to find the magnetic field strength (B), we can equate the two equations for F. Thus, we have:
(qvB) = (mv^2) / r
Simplifying the equation, we can cancel out the v term:
qB = (mv) / r
Now, let's plug in the given values:
Potential difference (V) = 4.7 kV = 4.7 × 10^3 V
Radius (r) = 8.0 cm = 8.0 × 10^-2 m
Charge of an electron (q) = -1.6 × 10^-19 C
Mass of an electron (m) = 9.1 × 10^-31 kg
We can calculate the velocity (v) by using the equation for potential difference and velocity:
V = Ed
Where:
V is the potential difference,
E is the electric field strength, and
d is the distance traveled.
Since the path is a circle, the distance traveled is equal to the circumference of the circle:
d = 2πr
Substituting the values:
4.7 × 10^3 V = E(2πr)
4.7 × 10^3 V = E(2π × 8.0 × 10^-2 m)
Solving for E:
E = (4.7 × 10^3 V) / (2π × 8.0 × 10^-2 m)
Now, we can calculate the velocity (v):
V = Ed
v = E × d
v = [(4.7 × 10^3 V) / (2π × 8.0 × 10^-2 m) ] × (2π × r)
v = [(4.7 × 10^3 V) / (8.0 × 10^-2 m) ] × (r)
Finally, we can substitute the values of q, m, v, and r into the equation qB = (mv) / r to find B:
(-1.6 × 10^-19 C)B = (9.1 × 10^-31 kg) × [(4.7 × 10^3 V) / (8.0 × 10^-2 m) ] × (8.0 × 10^-2 m)
Simplifying:
B = [(9.1 × 10^-31 kg) × (4.7 × 10^3 V)] / (-1.6 × 10^-19 C)
Evaluating the expression:
B = -1.3 × 10^-4 T
So, the strength of the magnetic field experienced by the electron is -1.3 × 10^-4 Tesla.