For, the central force F ( r ) , ∇ × F ( r ) = 0

Question

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Answer 1

Question

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

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Answer 2

Question

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

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Answer 3

Question

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

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Answer 4

Question

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

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Answer 5

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

Answer 6

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

Answer 7

Chapter 4, Problem 4.43P

(a)

To determine

For, the central force F(r), ∇×F(r)=0

Answer 8

(a)

Expert Solution

Answer to Problem 4.43P

For, the central force F(r), ∇×F(r)=0 is proved.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

The expression for Central force can be expressed as,

F(r)=f(r)r(xi^+yj^+zk^)=h(r)(xi^+yj^+zk^) (∵h(r)=f(r)r)

To avoid difficulty in calculation h(r) can be considered as f(r)r, h(r)=f(r)r.

Write the expression to calculate ∇×F(r)

∇×F(r)=|i^j^k^∂∂x∂∂y∂∂zh(r)xh(r)yh(r)z|=(z∂h(r)∂y−y∂h(r)∂z)i^+(x∂h(r)∂z−z∂h(r)∂x)j^+(y∂h(r)∂x−x∂h(r)∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^=(z(∂h(r)∂r∂r∂y)−y(∂h(r)∂r∂r∂z))i^+(x∂h(r)∂r∂r∂z−z∂h(r)∂r∂r∂x)j^+(y∂h(r)∂r∂r∂x−x∂h(r)∂r∂r∂y)z^ (I)

Write the expression magnitude of position vector,

r=x2+y2+z2

Differentiate with respect to x, y, and z.

∂r∂x=∂∂x(x2+y2+z2)=122x(x2+y2+z2)=x(x2+y2+z2)=xr

∂r∂y=∂∂y(x2+y2+z2)=122y(x2+y2+z2)=y(x2+y2+z2)=yr

∂r∂z=∂∂z(x2+y2+z2)=122z(x2+y2+z2)=z(x2+y2+z2)=zr

Conclusion:

Substitute xr for ∂r∂x, yr for ∂r∂y, and zr for ∂r∂z in (I) and solve.

∇×F(r)=(z(∂h(r)∂ryr)−y(∂h(r)∂rzr))i^+(x∂h(r)∂rzr−z∂h(r)∂rxr)j^+(y∂h(r)∂rxr−x∂h(r)∂ryr)z^=0

Therefore, conservative ∇×F(r)=0 is proved.

Answer 13

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

Answer 14

(b)

To determine

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

Answer 15

(b)

Expert Solution

Answer to Problem 4.43P

∇×F(r)=0 using the expression ∇×F(r) for polar is shown.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Use vector identity for ∇×F(r) in spherical polar coordinates.

∇×F(r)={r^1rsinθ(∂∂θ(sinθfϕ)−∂∂ϕfθ)+θ^(1rsinθ∂∂ϕfr−1r∂∂r(rfϕ))+ϕ^1r(∂∂r(rfϕ)−∂∂ϕfr)}=θ^1rsinθ∂∂ϕfr−ϕ^1r∂∂rfr=0

Conclusion:

Therefore, conservative ∇×F(r)=0 is proved much quicker using spherical polar corodinates.

Answer 20

Ch. 4 - Prob. 4.1P Ch. 4 - Prob. 4.2P Ch. 4 - Prob. 4.3P Ch. 4 - Prob. 4.4P Ch. 4 - Prob. 4.5P Ch. 4 - Prob. 4.6P Ch. 4 - Prob. 4.7P Ch. 4 - Prob. 4.8P Ch. 4 - Prob. 4.9P Ch. 4 - Prob. 4.10P

For, the central force F ( r ) , ∇ × F ( r ) = 0

Concept explainers

For, the central force F(r), ∇×F(r)=0

Answer to Problem 4.43P

Explanation of Solution

To show ∇×F(r)=0 using the expression ∇×F(r) for polar.

Answer to Problem 4.43P

Explanation of Solution

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