A 2.53-μC charged particle with a kinetic energy of 0.0929 J is fired into a uniform magnetic field of magnitude 0.147 T. If the particle moves in a circular path of radius 2.92 m, determine its mass.
centripetal force=magnetic force
mv^2/r=qvB
1/2 mv^2=rqvB/2
solve for v, you are given KE 1/2 mv^2, r, q, B
now knowing v, solve for mass
mass=2*KE/v
Well, well, well, it seems like we have a charged particle circus act going on here! Let's dive into this magnetic juggling act, shall we?
To determine the mass of our charged particle, we need to bring in the big guns: the equation for the centripetal force on a charged particle moving in a magnetic field!
The centripetal force is equal to the magnetic force acting on our charged particle. Mathematically, we have:
F_c = qvB
Where F_c is the centripetal force, q is the charge of the particle, v is its velocity, and B is the magnetic field strength.
Now, we know that the centripetal force can also be written as:
F_c = mv^2/r
Where m is the mass of the particle and r is the radius of the circular path.
Since both equations for the centripetal force must be equal to each other, we can equate them:
qvB = mv^2/r
Substituting the given values:
(2.53 x 10^-6 C)(v)(0.147 T) = m(v^2)/(2.92 m)
Now, let's simplify things a bit. The velocity of our particle cancels out, and we're left with:
(2.53 x 10^-6 C)(0.147 T) = (m)(v)
Now, we need to find the value of v, which we can calculate using our kinetic energy. The kinetic energy is given by:
K.E. = 1/2 mv^2
Rearranging the equation, we get:
mv^2 = 2K.E.
Plugging in the values:
(2.53 x 10^-6 C)(0.147 T) = (m)(2K.E.)
Now, we want to isolate the mass, so we divide both sides of the equation by 2K.E.:
m = (2.53 x 10^-6 C)(0.147 T) / (2K.E.)
Finally, we can substitute the given value of kinetic energy, 0.0929 J, into the equation:
m = (2.53 x 10^-6 C)(0.147 T) / (2(0.0929 J))
Now, let's grab our calculator and crunch those numbers to find our mass!
But remember, even though we're dealing with some serious calculations here, always keep a smile on your face like a clown would! Happy calculations!
To solve this problem, we can use the formula for the centripetal force experienced by a charged particle moving in a magnetic field:
F = (q * v * B) / R
Where:
F is the centripetal force
q is the charge of the particle
v is the velocity of the particle
B is the magnetic field magnitude
R is the radius of the circular path
We can rearrange the equation to solve for the velocity:
v = (F * R) / (q * B)
The kinetic energy of the particle also relates to its velocity:
KE = (1/2) * m * v^2
We can substitute the expression for v from the first equation into the second equation to find the mass (m):
m = (2 * KE) / v^2
Now, let's substitute the given values into these equations.
Given:
q = 2.53 μC = 2.53 * 10^-6 C
KE = 0.0929 J
B = 0.147 T
R = 2.92 m
Step 1: Calculate the velocity (v)
v = (F * R) / (q * B)
The force (F) acting as the centripetal force is provided by the kinetic energy:
F = KE / R
v = [(KE / R) * R] / (q * B)
v = KE / (q * B)
v = 0.0929 J / ((2.53 * 10^-6 C) * (0.147 T))
v ≈ 250.06 m/s
Step 2: Calculate the mass (m)
m = (2 * KE) / v^2
m = (2 * 0.0929 J) / (250.06 m/s)^2
m ≈ 2.35 * 10^-4 kg
Therefore, the mass of the charged particle is approximately 2.35 * 10^-4 kg.
To determine the mass of the particle, we can use the equation for circular motion in a magnetic field:
r = (m*v) / (|q| * B)
where:
r = radius of the circular path (given as 2.92 m)
m = mass of the particle (to be determined)
v = velocity of the particle
q = charge of the particle (given as 2.53 μC, which is equivalent to 2.53 x 10^-6 C)
B = magnitude of the magnetic field (given as 0.147 T)
First, we need to convert the charge from microcoulombs (μC) to Coulombs (C):
q = 2.53 x 10^-6 C
Next, we need to rearrange the equation to solve for the particle's velocity:
v = (r * |q| * B) / m
Rearranging the equation further to solve for the mass:
m = (r * |q| * B) / v
To find the velocity, we can use the relationship between kinetic energy (K) and velocity:
K = (1/2) * m * v^2
Rearranging the equation, we can solve for v:
v = sqrt((2 * K) / m)
Substituting this value of v into the equation for mass:
m = (r * |q| * B) / sqrt((2 * K) / m)
Now we can solve for the mass by rearranging the equation and squaring both sides:
m^2 = (r^2 * |q|^2 * B^2) / (2 * K)
m = sqrt((r^2 * |q|^2 * B^2) / (2 * K))
Finally, we can substitute the given values into the equation and calculate the mass.