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Determine the divergence of F = 30 i + 2xy j + 5xz2 k at (1,1,-0.2) and state the nature of the field.
- a)1, solenoidal
- b)0, solenoidal
- c)1, divergent
- d)0, divergent
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Determine the divergence of F = 30 i + 2xy j + 5xz2k at (1,1,-0.2) and...
Answer: b
Explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz2) = 0 + 2x + 10xz = 2x + 10xz
Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal.
Alternate/Shortcut: Without calculation, we can easily choose option b, as by theory when the divergence is zero, the vector is solenoidal. Option b is the only one which is satisfying this condition.
Explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz2) = 0 + 2x + 10xz = 2x + 10xz
Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal.
Alternate/Shortcut: Without calculation, we can easily choose option b, as by theory when the divergence is zero, the vector is solenoidal. Option b is the only one which is satisfying this condition.
Most Upvoted Answer
Determine the divergence of F = 30 i + 2xy j + 5xz2k at (1,1,-0.2) and...
Given F = 30 i 2xy j 5xz2k, we need to determine the divergence of the field at (1,1,-0.2) and state the nature of the field.
Divergence of F can be calculated using the formula:
div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
where Fx, Fy, and Fz are the x, y, and z components of F respectively.
Substituting the given values, we get:
div(F) = ∂(30)/∂x + ∂(2xy)/∂y + ∂(5xz2)/∂z
= 0 + 2x + 10xz
At (1,1,-0.2), we have:
div(F) = 2(1) + 10(1)(-0.2) = 0
Therefore, the divergence of the field is zero.
The nature of the field can be determined by analyzing the divergence. A zero divergence implies that the field is solenoidal, which means that the field lines form closed loops and there are no sources or sinks of flux. In other words, the field is a non-divergent, circulating flow.
Hence, the correct answer is option B: 0, solenoidal.
Divergence of F can be calculated using the formula:
div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
where Fx, Fy, and Fz are the x, y, and z components of F respectively.
Substituting the given values, we get:
div(F) = ∂(30)/∂x + ∂(2xy)/∂y + ∂(5xz2)/∂z
= 0 + 2x + 10xz
At (1,1,-0.2), we have:
div(F) = 2(1) + 10(1)(-0.2) = 0
Therefore, the divergence of the field is zero.
The nature of the field can be determined by analyzing the divergence. A zero divergence implies that the field is solenoidal, which means that the field lines form closed loops and there are no sources or sinks of flux. In other words, the field is a non-divergent, circulating flow.
Hence, the correct answer is option B: 0, solenoidal.
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Determine the divergence of F = 30 i + 2xy j + 5xz2k at (1,1,-0.2) and state the nature of the field.a)1, solenoidalb)0, solenoidalc)1, divergentd)0, divergentCorrect answer is option 'B'. Can you explain this answer?
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