zaz A proton of mass m is projected into a uniform field B = Ba₂ with an initial velocity aa + ßa₂. (a) Find the differential equations that the position vector r = xax + yay + must satisfy. (b) Show that a solution to these equations is -sin oot, y - cos wt, z = ßt where w = eB/m and e is the charge on the proton. (c) Show that this solution describes a circular helix in space. x dx Answer: (a)- = a cos wt, dt dy dt = -a sin wt, dz - = dt ß, (b) and (c) Proof.
zaz A proton of mass m is projected into a uniform field B = Ba₂ with an initial velocity aa + ßa₂. (a) Find the differential equations that the position vector r = xax + yay + must satisfy. (b) Show that a solution to these equations is -sin oot, y - cos wt, z = ßt where w = eB/m and e is the charge on the proton. (c) Show that this solution describes a circular helix in space. x dx Answer: (a)- = a cos wt, dt dy dt = -a sin wt, dz - = dt ß, (b) and (c) Proof.
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![A proton of mass \( m \) is projected into a uniform field \( \mathbf{B} = B_0 \mathbf{a_z} \) with an initial velocity \( \alpha \mathbf{a_x} + \beta \mathbf{a_z} \).
**(a)** Find the differential equations that the position vector \( \mathbf{r} = x \mathbf{a_x} + y \mathbf{a_y} + z \mathbf{a_z} \) must satisfy.
**(b)** Show that a solution to these equations is
\[
x = \frac{\alpha}{\omega} \sin \omega t, \quad y = \frac{\alpha}{\omega} \cos \omega t, \quad z = \beta t
\]
where \( \omega = \frac{eB_0}{m} \) and \( e \) is the charge on the proton.
**(c)** Show that this solution describes a circular helix in space.
**Answer:**
**(a)** The differential equations are:
\[
\frac{dx}{dt} = \alpha \cos \omega t, \quad \frac{dy}{dt} = -\alpha \sin \omega t, \quad \frac{dz}{dt} = \beta
\]
**(b)** and **(c)** Proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff26bd2c7-a69a-414f-95bc-e32ed97937d0%2Fa13b7698-6fe1-42a6-bca1-ac7e481abe7b%2Fqmq16w8_processed.png&w=3840&q=75)
Transcribed Image Text:A proton of mass \( m \) is projected into a uniform field \( \mathbf{B} = B_0 \mathbf{a_z} \) with an initial velocity \( \alpha \mathbf{a_x} + \beta \mathbf{a_z} \).
**(a)** Find the differential equations that the position vector \( \mathbf{r} = x \mathbf{a_x} + y \mathbf{a_y} + z \mathbf{a_z} \) must satisfy.
**(b)** Show that a solution to these equations is
\[
x = \frac{\alpha}{\omega} \sin \omega t, \quad y = \frac{\alpha}{\omega} \cos \omega t, \quad z = \beta t
\]
where \( \omega = \frac{eB_0}{m} \) and \( e \) is the charge on the proton.
**(c)** Show that this solution describes a circular helix in space.
**Answer:**
**(a)** The differential equations are:
\[
\frac{dx}{dt} = \alpha \cos \omega t, \quad \frac{dy}{dt} = -\alpha \sin \omega t, \quad \frac{dz}{dt} = \beta
\]
**(b)** and **(c)** Proof.
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