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Assume MKS units... Let Q be an open subset of R³. Let B: Q -R³ be a continuous vector .field, representing a magnetic field in 3-D spacc. Let p be a particle with charge q ER and mass m > 0. If p is at position 7=(x. y, z) in Q and E R³ is the velocity of P, at time t, then р feels a force ,) given by F(7₁7)=qx B(7). := " Suppose that p moves along a curve C as time t varies from a to b, and that p has position vector (t) and instantaneous velocity (t) at time t. (1) Explain why the two vectors (t) × (7(t)) and 7'(t) are perpen- dicular at every time t = [a, b]. (2) Using Part (1), calculate W = the work done on the particle p by the force as p moves from D = 7(a) to E = (b) along C. (3) Prove that d (|| T (t)||²³) = 2 T '(t) • F(t), dt at each time t. (4) Using Parts (2) and (3), and Newton's Second Law, prove that if the magnetic force (7,7) is the total force on p at every time t, then p moves along C at a constant speed.

Assume MKS units... Let Q be an open subset of R³. Let B: Q -R³ be a continuous vector .field, representing a magnetic field in 3-D spacc. Let p be a particle with charge q ER and mass m > 0. If p is at position 7=(x. y, z) in Q and E R³ is the velocity of P, at time t, then р feels a force ,) given by F(7₁7)=qx B(7). := " Suppose that p moves along a curve C as time t varies from a to b, and that p has position vector (t) and instantaneous velocity (t) at time t. (1) Explain why the two vectors (t) × (7(t)) and 7'(t) are perpen- dicular at every time t = [a, b]. (2) Using Part (1), calculate W = the work done on the particle p by the force as p moves from D = 7(a) to E = (b) along C. (3) Prove that d (|| T (t)||²³) = 2 T '(t) • F(t), dt at each time t. (4) Using Parts (2) and (3), and Newton's Second Law, prove that if the magnetic force (7,7) is the total force on p at every time t, then p moves along C at a constant speed.

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**Assume MKS units...** Let \( Q \) be an open subset of \( \mathbb{R}^3 \). Let \( \vec{B} : Q \rightarrow \mathbb{R}^3 \) be a continuous vector field, representing a magnetic field in 3-D space. Let \( p \) be a particle with charge \( q \in \mathbb{R} \) and mass \( m > 0 \). If \( p \) is at position \( \vec{r} = (x, y, z) \) in \( Q \) and \( \vec{v} \in \mathbb{R}^3 \) is the velocity of \( p \), at time \( t \), then \( p \) feels a force \( \vec{F}(\vec{r}, \vec{v}) \) given by \[ \vec{F}(\vec{r}, \vec{v}) := q \, \vec{v} \times \vec{B}(\vec{r}). \] Suppose that \( p \) moves along a curve \( C \) as time \( t \) varies from \( a \) to \( b \), and that \( p \) has position vector \( \vec{r}(t) \) and instantaneous velocity \( \vec{v}(t) \) at time \( t \). 1. **Explain why** the two vectors \( \vec{r}'(t) \times \vec{B}(\vec{r}(t)) \) and \( \vec{r}''(t) \) are perpendicular at every time \( t \in [a, b] \). 2. Using Part (1), **calculate** \( W \):= the work done on the particle \( p \) by the force \( \vec{F} \) as \( p \) moves from \( D = \vec{r}(a) \) to \( E = \vec{r}(b) \) along \( C \). 3. **Prove** that \[ \frac{d}{dt} \left( \|\vec{v}(t)\|^2 \right) = 2 \, \vec{v}'(t) \cdot \vec{v}(t), \] at each time \( t \
Transcribed Image Text:**Assume MKS units...** Let \( Q \) be an open subset of \( \mathbb{R}^3 \). Let \( \vec{B} : Q \rightarrow \mathbb{R}^3 \) be a continuous vector field, representing a magnetic field in 3-D space. Let \( p \) be a particle with charge \( q \in \mathbb{R} \) and mass \( m > 0 \). If \( p \) is at position \( \vec{r} = (x, y, z) \) in \( Q \) and \( \vec{v} \in \mathbb{R}^3 \) is the velocity of \( p \), at time \( t \), then \( p \) feels a force \( \vec{F}(\vec{r}, \vec{v}) \) given by \[ \vec{F}(\vec{r}, \vec{v}) := q \, \vec{v} \times \vec{B}(\vec{r}). \] Suppose that \( p \) moves along a curve \( C \) as time \( t \) varies from \( a \) to \( b \), and that \( p \) has position vector \( \vec{r}(t) \) and instantaneous velocity \( \vec{v}(t) \) at time \( t \). 1. **Explain why** the two vectors \( \vec{r}'(t) \times \vec{B}(\vec{r}(t)) \) and \( \vec{r}''(t) \) are perpendicular at every time \( t \in [a, b] \). 2. Using Part (1), **calculate** \( W \):= the work done on the particle \( p \) by the force \( \vec{F} \) as \( p \) moves from \( D = \vec{r}(a) \) to \( E = \vec{r}(b) \) along \( C \). 3. **Prove** that \[ \frac{d}{dt} \left( \|\vec{v}(t)\|^2 \right) = 2 \, \vec{v}'(t) \cdot \vec{v}(t), \] at each time \( t \
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