True or false: the curl of a vector field is itself a vector field.
Answer: C
It has a direction, given by the normal to the plane of greatest circulation, and a magnitude, the circulation per unit area. Divergence is the operation that returns a scalar, which is the distinction worth keeping firmly in mind.
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True or false: Stokes's theorem requires the surface of integration to be flat.
Answer: D
Any surface spanning the closed curve will do, flat or not, since the line integral round the boundary is the same for all of them. A closed surface, on the other hand, has no bounding curve at all, so the theorem then gives zero.
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Assertion: a static electric field can be written as the gradient of a scalar potential. Reason: the curl of a gradient vanishes identically.
Answer: A
The static field is irrotational, and the identity guarantees that any gradient is irrotational, so the representation is consistent. It is the vanishing curl of the static field that makes a scalar potential possible at all.
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Assertion: the magnetic flux density can be written as the curl of a vector potential. Reason: the divergence of a curl vanishes identically.
Answer: B
Since the flux density has zero divergence everywhere, it can consistently be represented as a curl, and the identity guarantees that any curl has zero divergence. The vector potential is not unique, since the gradient of any scalar may be added to it.
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Assertion: the volume element in spherical coordinates carries a factor \( r^2 \sin\theta \). Reason: the coordinate surfaces of the system are not planes.
Answer: C
The angular coordinates are not lengths, so each must be multiplied by the radius of the circle it sweeps to become one. That is a direct consequence of the curvilinear geometry, which the reason states in general terms.
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Match the operator with what it takes and returns.
1. Gradient 2. Divergence 3. Curl
P. Vector to scalar Q. Vector to vector R. Scalar to vector
Answer: D
Keeping these three straight prevents most of the errors in field theory: a divergence can never have a direction and a gradient can never be applied to a vector. It also explains at once why the Laplacian of a scalar is a scalar.
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Match the theorem or identity with its statement.
1. Divergence theorem 2. Stokes's theorem 3. Curl of a gradient
P. Zero for any scalar field Q. Volume integral to closed surface integral R. Surface integral to closed line integral
Answer: A
The two theorems each reduce an integral by one dimension, and each converts one of Maxwell's equations between differential and integral form. The identity is what allows a potential to be defined for an irrotational field.
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The steps of converting a Cartesian point to spherical coordinates are listed below. Put them in order.
1. Compute \( \theta = \arccos(z/r) \) 2. Compute \( r = \sqrt{x^2+y^2+z^2} \) 3. Check the quadrant of \( \phi \) from the signs of x and y 4. Compute \( \phi = \arctan(y/x) \)
Answer: B
The radius must come first because the polar angle is defined through it. The azimuthal angle follows, and the quadrant check is essential because the arctangent alone cannot distinguish opposite directions in the plane.
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Consider the following statements about the del operator.
1. The curl of a gradient is zero 2. The divergence of a curl is zero 3. The gradient of a divergence is zero
Answer: C
The first two are genuine identities and both are used constantly in field theory. The third is not an identity at all: the gradient of a divergence is a perfectly good non-zero vector, and it appears in the expansion of the curl of a curl.
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Consider the following statements about coordinate systems.
1. The cylindrical system suits a long straight conductor 2. The spherical system suits a point charge 3. The Cartesian unit vectors vary with position
Answer: D
Choosing the system whose symmetry matches the problem is what makes a field calculation tractable. The Cartesian unit vectors are the only ones that do not vary with position, which is precisely why the other two systems have more complicated operator forms.
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