152 Chapter Three (a) Show that the acceleration of the charge is given by az = 9E (1 - 12) 3/2 mo (b) Show that the velocity of the charge at any time t is given by qEt/mo U₂ = V1+ (qEt/moc)2 (c) Show that the distance the charge moves in a time t is given by x = moc² qE (√1 + (qEt/moc)² - 1). (d) Show that when t is large, uz approaches c, and x approaches ct. (e) Show that if qEt/moc we obtain the classical results for az, Ur, and x. RELATIVISTIC DYNAMICS >
152 Chapter Three (a) Show that the acceleration of the charge is given by az = 9E (1 - 12) 3/2 mo (b) Show that the velocity of the charge at any time t is given by qEt/mo U₂ = V1+ (qEt/moc)2 (c) Show that the distance the charge moves in a time t is given by x = moc² qE (√1 + (qEt/moc)² - 1). (d) Show that when t is large, uz approaches c, and x approaches ct. (e) Show that if qEt/moc we obtain the classical results for az, Ur, and x. RELATIVISTIC DYNAMICS >
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![**Chapter Three: Relativistic Dynamics**
(a) **Show that the acceleration of the charge is given by:**
\[a_x = \frac{qE}{m_0} \left( 1 - \frac{u^2}{c^2} \right)^{3/2}\]
(b) **Show that the velocity of the charge at any time \(t\) is given by:**
\[u_x = \frac{qEt/m_0}{\sqrt{1 + \left( qEt/m_0c \right)^2}}\]
(c) **Show that the distance the charge moves in a time \(t\) is given by:**
\[x = \frac{m_0 c^2}{qE} \left( \sqrt{1 + \left( qEt/m_0c \right)^2} - 1 \right)\]
(d) **Show that when \( t \) is large, \( u_x \) approaches \( c \), and \( x \) approaches \( ct \).**
(e) **Show that if \( qEt/m_0 \ll c \) we obtain the classical results for \( a_x \), \( u_x \), and \( x \).**
In these equations:
- \( q \) denotes the charge.
- \( E \) represents the electric field strength.
- \( m_0 \) is the rest mass of the charge.
- \( u \) is the velocity of the charge.
- \( c \) stands for the speed of light in a vacuum.
- \( t \) indicates time.
- \( a_x \) is the acceleration of the charge in the x-direction.
- \( u_x \) is the velocity of the charge in the x-direction.
- \( x \) represents the distance moved by the charge in the x-direction.
These equations describe the motion of a charged particle in an electric field under relativistic conditions, highlighting how quantities like acceleration, velocity, and displacement differ from classical mechanics as the influence of the speed of light becomes non-negligible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4e83917-f32f-4dfc-9868-9fc8593391ec%2F88eb906d-e8e4-49f8-8315-adfe86690c10%2F6a6n4i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Chapter Three: Relativistic Dynamics**
(a) **Show that the acceleration of the charge is given by:**
\[a_x = \frac{qE}{m_0} \left( 1 - \frac{u^2}{c^2} \right)^{3/2}\]
(b) **Show that the velocity of the charge at any time \(t\) is given by:**
\[u_x = \frac{qEt/m_0}{\sqrt{1 + \left( qEt/m_0c \right)^2}}\]
(c) **Show that the distance the charge moves in a time \(t\) is given by:**
\[x = \frac{m_0 c^2}{qE} \left( \sqrt{1 + \left( qEt/m_0c \right)^2} - 1 \right)\]
(d) **Show that when \( t \) is large, \( u_x \) approaches \( c \), and \( x \) approaches \( ct \).**
(e) **Show that if \( qEt/m_0 \ll c \) we obtain the classical results for \( a_x \), \( u_x \), and \( x \).**
In these equations:
- \( q \) denotes the charge.
- \( E \) represents the electric field strength.
- \( m_0 \) is the rest mass of the charge.
- \( u \) is the velocity of the charge.
- \( c \) stands for the speed of light in a vacuum.
- \( t \) indicates time.
- \( a_x \) is the acceleration of the charge in the x-direction.
- \( u_x \) is the velocity of the charge in the x-direction.
- \( x \) represents the distance moved by the charge in the x-direction.
These equations describe the motion of a charged particle in an electric field under relativistic conditions, highlighting how quantities like acceleration, velocity, and displacement differ from classical mechanics as the influence of the speed of light becomes non-negligible.
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