The flux of E from a cylinder with uniform ρ inside?

Depends on radius & enclosed charge

The Gauss’s law boundary condition for E normal component is?

E⊥ above = E⊥ below + σ/ε0

The potential from multiple charges is?

Sum of each charge’s potential

A ring of uniform charge λ. E on axis is?

E= (1/4πε0)(q z / (z²+R²)^(3/2))

Helmholtz theorem demands knowledge of?

∇·F & ∇×F + boundary conditions

The radial part of ∇²f in spherical coords?

(1/r²)(∂/∂r)(r² ∂f/∂r)

The Laplacian of a scalar in Cartesian coords?

∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²

The gradient in spherical coords is?

∂T/∂r r̂ + 1/(r sinθ) ∂T/∂φ φ̂ + ...

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Questions 174

Answers

1
The boundary condition for potential across a surface with no free charge?
V is continuous
V must vanish
No condition known
V jumps by ρ/ε0
2
The flux of E from a cylinder with uniform ρ inside?
Always 0
Depends on radius & enclosed charge
Infinite for large radius
No direct formula
3
The condition E=−∇V demands?
∇×E=0
E=∞ at origin
No boundary conditions
∇·E=0
4
The Gauss’s law boundary condition for E normal component is?
E⊥ above = E⊥ below + σ/ε0
∇×E=0 implies no jump
No net difference
E⊥ is always continuous
5
A field E that is purely radial and ∝1/r² has?
Zero curl except at r=0
Zero div everywhere
No physical meaning
Infinite flux everywhere
6
The boundary condition for E⊥ at a surface?
E⊥ changes by surface current
E⊥ changes by ρ
E⊥ changes by σ/ε0
E⊥ is continuous always
7
The potential from multiple charges is?
Infinite always
No superposition
Needs advanced PDE solver
Sum of each charge’s potential
8
For a uniform volume charge ρ, total charge in sphere radius R?
(4/3)πR³ρ
No finite sum
ρR²
4πR²ρ
9
A ring of uniform charge λ. E on axis is?
(1/4πε0)(q / z²)
(σ/2ε0)ẑ
E= (1/4πε0)(q z / (z²+R²)^(3/2))
Zero at any z
10
If ∂f/∂x=0, then f=?
No dependence on any var
Constant in x direction
Must be zero
No conclusion
11
Helmholtz theorem demands knowledge of?
Only F at one boundary
Just ∇·F alone
∇·F & ∇×F + boundary conditions
Time derivatives of F
12
The direction of E for a uniform line charge?
Circles around wire
Radial from wire
Along wire
Zero inside
13
The field E near an infinite plane with charge σ is?
σ/ε0
σ/2ε0
No field
Depends on distance
14
The quantity ∇·(r̂/r²)=?
Constant=4π
∞ at r=∞
No meaning
0 for r≠0, 4πδ3(r) at r=0
15
The line integral ∫E·dl from r to ∞ is?
+V(r)
No significance
∞ always
−V(r) if V(∞)=0
16
A necessary condition for E to be electrostatic?
∇·E=0
∇·E=ρ/ε0
∇²E=0
∇×E=0
17
A volume charge density ρ(r)=k r means?
Impossible physically
Spherical shell only
No net charge
Charge grows with radius
18
E of a charged conducting sphere acts?
Only tangential
No simple form
Like point charge at center
Zero outside
19
In cylindrical coords, ŝ·φ̂=?
No constant value
1
0
s
20
If E=−∇V, then the line integral ∫E·dl=?
0 always
−(V(b)−V(a))
(V(b)+V(a))
Undefined
21
Which is correct for ∇×(∇×A)?
∇(∇·A) − ∇²A
∇(∇²A) − ∇·A
∇²(∇·A) + A²
0 always
22
A closed surface integral of E·da=0 implies?
No charges anywhere
∇×E=0
Charge is on surface only
No net enclosed charge
23
In electrostatics, “equipotential” means?
Surface where E=0
Surface of constant V
Any boundary of a region
Surface of zero charge
24
A field with zero divergence is called?
Solenoidal
Irrotational
Discontinuous
Conservative
25
The boundary condition for E∥ at a surface?
E∥ is always continuous
E∥ changes by σ/ε0
E∥ is always zero
No known condition
26
If E has zero curl, it can be written as?
E=∞ at boundary
E=∇×A
No expression
E=−∇V
27
For E = (x^2) x̂ + 2xy ŷ + 0 ẑ, compute ∇·E.
4xy
2x+2y
2x + 2x=4x
2x
28
The radial part of ∇²f in spherical coords?
No known expression
(1/r)(∂/∂r)(r ∂f/∂r)
(1/r²)(∂/∂r)(r² ∂f/∂r)
(∂²f/∂θ²)
29
If E=−∇V, then V is called?
An equipotential
A current density
A vector potential
The scalar potential
30
The Maxwell equation ∇×E=−∂B/∂t reduces to ∇×E=0 when?
B=0
Static case (∂B/∂t=0)
No charges
Infinite conductivity
31
The area vector for a curved surface is?
Points in direction of E
Always normal to surface
Tangential to surface
No direction assigned
32
In cylindrical coords, volume element dτ=?
s ds dφ dz
r² dr dθ dz
(1/r)dr dθ dz
r dr dθ dz
33
The step function θ(x) derivative is?
1
∞
δ(x)
0
34
A set of surfaces with constant V are?
Equipotentials
No meaning in electrostatics
Flux lines
Solenoidal surfaces
35
The potential at center of uniformly charged sphere is?
(3/2)(Q/4πε0R)
Q/4πε0R
∞
Zero
36
If ∇²V=0 in some volume, V is said to be?
Divergent
No name
Dipole-like
Harmonic in that volume
37
The identity ∇(f g)=f ∇g+g ∇f is?
A sum rule
A product rule
A chain rule
No name
38
∇·(1/r) in 3D is?
∞ at r=0, 0 elsewhere
1/r²
4πδ³(r)
Unknown
39
Gauss’s law says net flux out of volume=?
No direct meaning
Charge total / µ0
Charge inside / ε0
0 if ∇×E=0
40
If ρ=0 in some region, ∇²V=0. This is?
Helmholtz eqn
Poisson’s eqn
No name
Laplace’s eqn
41
The electric potential of a dipole (p) at large r?
No potential for dipoles
V=∞
V~p·r̂ / (4πε0 r²)
V~p / r
42
The Laplacian of a scalar in Cartesian coords?
∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²
∇²f=∂²f/∂r² only
∇²f = ∂f/∂x + ...
None are correct
43
E for a spherical shell inside (r
Constant
∞
Zero
q/r²
44
A closed surface that encloses no charge has?
Flux=0
Flux=∞
Flux=nonzero
Flux depends on shape
45
If ∇²V=0, V obeys Laplace’s eqn. Solutions are?
Unique with boundary conds
Multiple solutions always
Infinite set random
No solutions exist
46
The vector identity ∇×(A×B)=?
∇(AB)
(B·∇)A − (A·∇)B
Always zero
No standard formula
47
In Griffiths’s notation, r = ?
Any separation vector to field
(x−x')x̂ + ... field point
x x̂ + y ŷ + z ẑ from source
Position vector from origin
48
The gradient in spherical coords is?
∂T/∂r r̂ + 1/(r sinθ) ∂T/∂φ φ̂ + ...
∂/∂r r̂ + ∂/∂θ θ̂ + ...
Use chain rule in x,y,z only
∂T/∂r r̂ + (1/r)∂T/∂θ θ̂ + ...
49
A delta function in 3D is δ3(r). Integral=?
Zero unless r=∞
1 over entire space
∞
1 at r=0, else 0
50
In Griffiths, “r” typically denotes?
None of these
x̂ + ŷ + ẑ
A unit vector in x
Position vector from origin
51
For E to be continuous tangentially across a boundary, we require?
No surface charge
No jump in V
No net flux
No current flow
52
E from infinite line charge is found by?
Integrating Coulomb’s law or Gauss’s law
Stokes’ theorem only
Amperian loop method
No method works
53
The integral ∫(∇×A)·da=?
∮A·dl
∫A·da
No known formula
0 always
54
∇2(1/r)=?
Always 0
∞ for all r
1/r²
0 for r≠0, −4πδ3(r)
55
If a function has constant gradient, the function is?
No solutions
Quadratic in coords
Divergent at origin
Linear in coordinates
56
If ∇·E=4πδ(r) in CGS, then in SI?
∇×E=0
∇·E=0
No direct conversion
∇·E=ρ/ε0
57
A uniform E field can be derived from?
Surface integral approach
Constant potential gradient
Dipole array only
No potential possible
58
The integral of δ3(r−r') over all space?
1
∞
0 always
|r−r'|²
59
The dipole moment p=?
q d in direction from − to +
Sum of charges
No standard definition
q (r1−r2) always
60
The condition for a static electric field in matter?
B is always present
∂E/∂t=∞
Charges are moving fast
E=0 in conductor interior
61
The Dirichlet boundary condition for V?
Specifies V on the boundary
Specifies no charges
Specifies conduction
Specifies E normal comp
62
If ∇²V=−ρ/ε0, then for ρ=0 we get?
Continuity eqn
∇²V=0 Laplace eqn
No eqn
∇·E=∞
63
Divergence theorem: ∫V(∇·A)dτ=?
∫V(∂A/∂t)dτ
∮S(A×dl)
∮S(A×da)
∮S(A·da)
64
The integral ∫∞→0 of e−x dx=?
-1
1
0
∞
65
The flux of E through a surface measures?
Number of field lines crossing
∫E×da
Surface tension
Energy stored in E
66
A function with zero Laplacian is?
An eigenfunction
No significance
Called harmonic
Divergent at origin
67
Gauss’s law for a point charge q at origin?
∮E·da = 0
∮E·da = q²/r²
∮E·da = q/ε0
∮E·da = 4πρ
68
For a rectangular region with uniform ρ, E at corners?
No direct formula
Always zero
Varies, but found by symmetry
Must be the same
69
The function r̂ in spherical coords depends on?
x,y,z linearly
No angles needed
θ, φ
Only r
70
E=−∇V means E is?
Only radial
Non-conservative
Complex vector
Conservative field
71
The function V that solves Laplace’s eqn in a region is?
Infinite at boundary
Always zero inside
Harmonic there
Unknown
72
The length element in cylindrical coords?
dr r̂ + r dθ θ̂ + dz ẑ
ds ŝ + s dφ φ̂ + dz ẑ
No expression
s ds dφ dz
73
The dot product of (x̂+ŷ) with (x̂−ŷ)?
1−1=0
2
1+1=2
-2
74
The Neumann boundary condition for V?
Sets V=0 on boundary
No name in math
Specifies ∂V/∂n on boundary
No flux on boundary
75
The potential of an infinite line charge?
No potential is possible
~λ/r²
~2λ ln(r)/4πε0
No finite form
76
E field lines never do what?
Leave a negative charge
Cross each other
Extend to infinity
Begin or end on charges
77
The vector identity for ∇·(φA) is?
φ(∇·A)+A·(∇φ)
(A·∇)φ + ...
∇×(φA)
(∇φ)×A
78
The integral ∮(r̂/r²)·da over a closed surface not containing r=0 is?
0
∞
No definite value
4π
79
In spherical coords, ∇·v = ?
No known formula
(1/r²)(∂/∂r)(r² vr) + ...
(∂/∂x)(vx)+(∂/∂y)(vy)+...
(1/r)(∂/∂r)(r vr) + ...
80
The partial derivative ∂/∂x means?
Hold y,z constant
Hold time constant
Take limit r→0
No constraints
81
The potential due to a point charge q at r' is?
V(r)=−(q/ε0)r
V(r)=q/(4πε0) · 1/r
V(r)=q/r²
V(r)=q E(r)
82
The cross product of two gradient fields?
Always zero
Infinite at origin
Depends on boundary
Never zero
83
Divergence of r̂/r² in 3D is?
Always 0 everywhere
Constant=4π
∞ everywhere
0 for r≠0, 4πδ³(r) at r=0
84
The vector potential A is used mostly for?
Fluid dynamics only
Heat conduction
Electrostatics
Magnetostatics
85
The integral ∮(∇×F)·da=?
0
∮F·dl
∞
No expression
86
The 2D version of ∇×(∇φ)=0 means?
It’s a guess
φ is indefinite
No direct meaning
∂²φ/∂x∂y=∂²φ/∂y∂x
87
A line integral ∫F·dl from r=a to r=b is path-independent if?
No conditions
F is constant
∇×F=0
∇·F=0
88
For ∇²(1/r)=?,
−4πδ3(r)
No solution
1/r²
0 everywhere
89
The fundamental theorem of calculus in 3D?
Is ∇·E=ρ/ε0
∇²=0
Refers to line integrals of grad
∇×E=0 always
90
If ∫E·dl=0 for all loops, then?
E has zero divergence
E=−∇V for some V
E is a perfect conductor
E must vanish
91
In cylindrical coords, ∇2f=?
(∂f/∂x)+(∂f/∂y)+...
1/s ∂/∂s(s ∂f/∂s)+...
(∂²f/∂r²)+(1/r)∂f/∂r + ...
None
92
For a rectangular box, flux of E depends on?
Size of box only
Charge inside only
Boundary conditions on V
Surface orientation
93
The flux of r through a sphere radius R?
=3(4πR³/3)=4πR³
4πR³, no factor
∫r·da=3Volume
=r²? No
94
The indefinite integral of δ(x) is?
No closed form
ln|x|
Zero always
Heaviside step function
95
The triple product A·(B×C) is?
Always 0
(A×C)·B
A·B + B·C
(A×B)·C
96
Divergence in spherical coords includes factor?
1/r
1/r²
No factors
1/s
97
If ∇·E=ρ/ε0, that means physically?
No significance
No boundary conditions needed
Always zero flux
Charges create divergence of E
98
The indefinite integral of sin(x)dx=?
cos(x)+C
∞
−cos(x)+C
No elementary form
99
The line integral ∮E·dl=0 means E is what?
Harmonic
Solenoidal
Conservative
Divergence-free
100
The fundamental theorem for div: ∫(∇·F)dτ=?
∫(F·da)
∫(F×dl)
∮(F×da)
∮(F·dl) around loop
101
The line integral of E from a to b is?
Dipole moment
Flux of E
Always zero
Potential difference V(b)-V(a)
102
The electric field inside a conductor in electrostatics?
Constant but nonzero
Infinite
Zero in static equilibrium
Nonzero if charges exist
103
The derivative ∂/∂z of x²+3y²+2z²=?
2z²
4z
4z²
4z + 0
104
A physically allowed electrostatic field must have?
∇×E≠0
∇×E=0
∇·E=0 always
No boundary conditions
105
If a field has zero curl, it is called?
Irrotational
Solenoidal
Radial only
Harmonic
106
The integral ∫∞→0 (1/x²) dx=?
Diverges
1
1/∞=0
No meaning
107
If E= kx x̂, the flux over cube center at origin?
Zero net flux
Infinite
Nonzero flux
Need advanced integration
108
If ∇×E=0, what does that say about E?
E can be written as −∇V
E points in circles
E is always constant
E must vanish
109
The integral of δ(x−a)f(x) dx=?
∞ unless x=a
f(x) for all x
f(a)
0 always
110
The vector area a of a flat surface is?
Integral of x̂+ ŷ + ẑ
Zero always
Magnitude=area, direction=normal
The perimeter × normal
111
Which vector identity implies ∇×(∇V)=0?
Gauss’s theorem
Curl of gradient is zero
Helmholtz decomposition
d/dt=0 identity
112
If E is radial ∝1/r², flux over sphere is?
Diverges with r→∞
= 0 always
Constant, = q/ε0
∝ r²
113
Electric field lines begin on ___ charges.
Either sign
Negative
Positive
They don't start on charges
114
The flux of E for a dipole over large sphere?
Zero
=p/ε0
No unique value
∞
115
The flux of ∇×F over any closed surface is?
Nonzero if F=∇φ
Undefined
=4π times the charge
Always zero
116
A unit normal to a plane x=2 is?
x̂
No unique direction
−x̂
2x̂
117
The unit vector in direction of increasing φ in spherical coords?
φ̂=−sinφ x̂+cosφ ŷ
No direct formula
φ̂=−sinθ cosφ x̂...
φ̂=cosφ x̂+sinφ ŷ
118
In spherical coords, r̂ · θ̂ = ?
sinθ
0
cosθ
1
119
The dot product (r̂)·(θ̂)=?
sinθ
0
r
1
120
∇²V=−ρ/ε0 is known as?
Coulomb’s eqn
Poisson’s eqn
Faraday’s eqn
Green’s identity
121
If E=0 in a region, that means?
All charges are outside
No charges in that region
Potential is constant there
All are correct
122
If ∇·F=0, then F can be written as?
F=0 always
F=−∇V
F must be radial only
F=∇×A
123
A boundary where conductor meets vacuum means?
Surface charge might appear
Always uniform E
No field difference
E=0 in vacuum
124
A function with ∇²f=0 in a region is?
Divergent
Harmonic in that region
Dipolar always
Poisson function
125
The expression ∫(∇·E)dτ= Qenc/ε0 is?
Poisson eqn result
Continuity eqn
Divergence theorem + Gauss’s law
Stokes’ theorem result
126
The direction of ∇f is?
Max increase of f
No direction
Min increase
Always radial
127
In spherical coords, the Laplacian of f(r)?
∂²f/∂θ² + ... only
(1/r)∂/∂r(r ∂f/∂r)+...
(1/r²) d/dr[r² df/dr]
No known expression
128
The integral of x dx from 0 to a?
No solution
1/2
∞
a²/2
129
In spherical coords, da for a sphere radius R=?
R² sinθ dθ dφ r̂
No formula needed
R² dθ dφ r̂ + ...
R dr dφ θ̂
130
The radial unit vector r̂ in spherical coords is?
(sinφ)x̂ + ...
(sinθ cosφ)x̂ + ...
(cosθ)x̂ + ...
(−sinφ)x̂ + ...
131
For a uniformly charged sphere, E outside?
Depends on shape of Gaussian surface
Acts like a point charge q at center
Zero everywhere
Infinite at r=R
132
The potential for a uniform sphere of charge outside the sphere is?
Like a point charge Q
Zero outside
Infinite for r>R
No expression exists
133
The boundary condition for E at conductor surface with σ?
E∥=0
No condition known
E=∞
E⊥=σ/ε0
134
For an infinite plane with charge σ, E=?
σ/ε0 (one side)
σ/2ε0 (both sides)
σ/(4πε0r²)
0 outside, σ/2ε0 inside
135
For an infinite cylinder, Gauss’s law uses?
A coaxial cylindrical surface
A spherical surface
A rectangular surface
A pillbox shape
136
For a uniform plane of charge, E does not depend on distance because?
No 1/r² drop, plane is infinite
Surface is conductor
It actually does drop
Trick question
137
For a uniform line charge λ, infinite wire, E=?
E=λ/(2πε0 r)
E=λ/(4πε0 r²)
No finite value
E=σ/(ε0)
138
The conduction electrons in equilibrium?
Make net E=0 inside conductor
Cause B fields
Flow constantly
Generate infinite potentials
139
Dirac delta δ(x) has what property?
Value=1 for x=0
∞ at x=0, integral=1
It's a constant function
Always 0, integral=∞
140
The integral ∫(∇²V)dτ can be transformed into?
∮(∇V·da) by the divergence theorem
No known transform
∮(V dl)
∮(E·da)
141
A vector field with ∇·F=ρ/ε0 and ∇×F=0 might be?
None
E field in electrostatics
B field in magnetostatics
Velocity field of fluid
142
For a large neutral sphere next to a point charge?
Infinite potential
Dipole only
No net field
Field is superposition sphere+point
143
In an electrostatic conductor, net charge resides?
Throughout the volume
No charge at all
On the surface
At center only
144
The potential at infinity is usually set to?
No standard choice
∞
q/4πε0∞
Zero
145
The path integral of a gradient field?
Is zero for all endpoints
Is path-independent
Depends on path
Always negative
146
The integral ∫(∂ρ/∂t)dτ=−∫(J·da) is?
Continuity eqn
Ampère’s law
Gauss’s law
Kirchhoff’s rule
147
A PDE form: ∇²V=−ρ/ε0 is known as?
Poisson’s equation
Gauss’s eqn
Maxwell’s eqn
Laplace’s eqn
148
A gradient field’s curl is always?
Nonzero at r=0
Zero
No simple rule
Infinite on surfaces
149
In cylindrical coords, dl = ?
dr r̂ + r dθ θ̂ + dz ẑ
None of these
ds ŝ + s dφ φ̂ + dz ẑ
dx x̂ + dy ŷ + dz ẑ
150
The line element in spherical coords is dl=?
dx x̂ + dy ŷ + dz ẑ
ds ŝ + s dφ φ̂ + dz ẑ
No expression
dr r̂ + r dθ θ̂ + r sinθ dφ φ̂
151
The partial derivatives of V are used to get E how?
E=−∂V/∂t
E=∇×V
No direct way
E=−∇V
152
If ∫E·dl=0 around all loops, E must be?
Zero at boundaries
Divergent
Constant
A gradient field
153
The unit vector φ̂ in cylindrical coords is?
(−sinφ) x̂ + (cosφ) ŷ
(cosφ) x̂ + (sinφ) ŷ
ẑ
(sinφ) x̂ − (cosφ) ŷ
154
The area element on a sphere radius R is?
R² dφ dθ cosθ
2πR dθ
No formula known
R² sinθ dθ dφ
155
The integral ∮(r̂/r²)·da over a sphere=?
4π
∞
0
1
156
Divergence of r is?
0
3
r²
1
157
A potential that depends only on r in spherical coords suggests?
No spherical symmetry
Spherical symmetry of E
Always a dipole
E is zero inside
158
The line integral ∫0→π of sinθ dθ=?
1
2
0
-2
159
The indefinite integral of 1/r² dr=?
∞
−1/r
r²/2
ln(r)
160
Gauss’s law in integral form is ∮E·da=?
μ0I_enc
0
−dΦB/dt
Q_enc / ε0
161
The boundary condition for V across a surface?
V jumps by ρ/ε0
No condition known
V is continuous
V must vanish at boundary
162
The line integral ∫E·dl from a to b is?
Dipole moment
Potential difference V(b)-V(a)
Flux through a surface
Always zero
163
The Helmholtz theorem states that a vector field is fully known if we know?
Only its magnitude
Its gradient alone
Nothing else needed
Its div & curl + boundary
164
The region where ∇²V=−ρ/ε0 includes?
Everywhere with ρ>0
No boundary
Inside conductors only
No solutions
165
The expression (r̂/r²) is derived from?
No known origin
Gradient of 1/r with sign
Constant vector
Curl of 1/r
166
The partial derivative ∂(xyz)/∂x=?
yz
xy
0
x+y+z
167
The difference V(b)−V(a)=−∫(E·dl) implies?
Path independence if ∇×E=0
No physical meaning
Multiple values for V
Path dependence always
168
The sum of partial derivatives in ∇·E is?
∂Ex/∂x+∂Ey/∂y+∂Ez/∂z
Ex+Ey+Ez
∞ at origin
Always zero in electrostatics
169
“r = r−r'” in Griffiths is?
Radius vector in cylinder
Separation vector field→source
Points from field to origin
No meaning
170
∫(∇×E)·da over any surface=?
0 if electrostatic
∞ if charges inside
No known result
∮E·dl around boundary
171
The integral ∫(∇·E)dτ= ?
∮(E·da)
∫(∂E/∂t)dτ
∇×E=0
−∫(E·dl)
172
What does ∇·E=ρ/ε0 represent?
Ampère’s law with no current
Gauss’s law in differential form
Boundary condition for E
Faraday’s law of induction
173
For a rectangular box with uniform ρ, the total charge is?
No real meaning
ρ×Volume
∞
Depends on shape of surface
174
The fundamental theorem of gradients says?
∇²T=0 inside conductor
No known statement
∮(∇T)·dl= Q/ε0
∫(∇T)·dl = T(b)−T(a)

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