The physical law that electric charge is conserved is expressed as continuity equation ∂ρ/∂t + ∇·J = 0. This expresses local conservation of __________.
A mass
B charge
C energy
D momentum
ρ is charge density, J current density — continuity expresses charge conservation.
In spherical coordinates volume element dV = r^2 sinθ dr dθ dφ. The factor r^2 sinθ is the __________.
A Jacobian determinant
B density
C gravitational factor
D error term
r^2 sinθ arises from Jacobian of transformation.
Which of the following is a correct statement of Noether’s theorem (simplified)?
A Every symmetry leads to a conserved quantity
B Symmetries do nothing
C Only parity gives conservation
D No relation between symmetry and conservation
Noether links continuous symmetries to conservation laws.
The SI derived unit for work is __________.
A Newton
B Joule
C Watt
D Pascal
Work/energy measured in Joules (N·m).
The scalar product of two vectors with an angle of 60° and magnitudes 4 and 3 equals __________.
A 12
B 6
C 4
D 0
4·3·cos60° = 12*(0.5) = 6.
A measurement reported as 0.00340 has how many significant figures?
A 3
B 4
C 2
D 5
0.00340 has significant digits 3,4,0 → three significant figures.
The relation F = −∇V indicates force is __________ of potential V.
A negative gradient
B positive gradient
C curl
D divergence
Conservative force equals negative gradient of potential.
The perimeter of a circle does not involve solid angle because solid angle is a __________.
A 3D angular measure
B linear measure
C time measure
D mass measure
Solid angle is 3D; perimeter concerns 2D.
Which of these is a vector differential operator?
A gradient (∇)
B Laplacian (∇²) is scalar operator
C both A and B
D none
∇ is vector operator; ∇² is scalar operator derived from ∇·∇.
The magnitude of angular momentum L = r × p is same as __________.
A rp sinθ p?
B r p sinθ
C r+p
D r·p
|L| = r p sinθ where θ is angle between r and p.
In error propagation, the uncertainty of a sum z = x + y (independent) is given by __________.
A Δz = Δx + Δy
B Δz = sqrt(Δx^2 + Δy^2) (for random independent errors)
C Δz = Δx − Δy
D Δz = Δx × Δy
For independent random errors, add in quadrature.
Which of the following is NOT required for a coordinate system?
A origin
B axes or reference directions
C units of measure
D gravity
Gravity is not required for coordinate definition.
The rotational symmetry group of a square in plane is of order __________.
A 1
B 2
C 4
D infinite
Square has four rotations (0°,90°,180°,270°) forming C4 group.
The derivative d(êθ)/dθ equals __________ (in polar coordinates).
A −êr
B êr
C zero
D êθ
Unit tangential vector derivative yields −radial unit vector: dêθ/dθ = −êr.
Which of the following gives units of force density (force per unit volume)?
A N/m^3
B N/m^2
C N·m
D kg·m/s
Force per unit volume units N/m^3.
The right-hand rule helps determine direction of __________.
A cross product
B dot product
C scalar addition
D derivative sign
Right-hand rule indicates direction of a×b.
The equation pV = nRT is dimensionally consistent because both sides have dimensions of __________.
A pressure × volume = energy
B temperature only
C force
D momentum
pV has units of energy; RHS nRT also energy-related.
The concept of reference frame is important because physical quantities like velocity are __________.
A frame-dependent
B frame-independent always
C imaginary
D constant in all frames
Velocity depends on chosen reference frame.
Which combination yields a quantity with dimensions of velocity?
A length × time
B length / time
C mass/time
D force × time
Velocity = length/time.
The operation to find the solid angle subtended by a surface at a point often requires integrating __________.
A dA/r^2 projected on sphere
B force per area
C length element only
D temperature gradient
Integrate dΩ = (dA cosθ)/r^2 or area/r^2 depending on geometry.
The bias in measurement that shifts results in one direction is called __________.
A systematic error
B random error
C statistical scatter
D noise only
Systematic bias shifts measurements consistently.
In units, 1 Pa = 1 N/m^2 = 1 kg·m^-1·s^-2. True or False?
A True
B False
C Only in CGS
D Depends on gravity
Pa is defined as N/m^2 → kg·m^-1·s^-2.
The time reversal operator T changes sign of __________.
A position
B velocity and momentum
C energy
D mass
Under t→−t velocities and momenta change sign; positions unaffected.
If you rotate a coordinate system, tensor components transform according to __________.
A same numbers always
B representation rules depending on rank (matrix multiplication by rotation matrices)
Which of the following is a correct statement: “Solid angle is measured in steradians which are __________.”
A an SI derived unit and dimensionless
B base unit of SI
C not used in physics
D has dimensions of [L^2]
Steradian is SI derived dimensionless unit.
The expectation that physical constants are the same in all inertial frames is part of __________.
A principle of relativity
B quantum mechanics only
C statistical mechanics only
D thermodynamics only
Principle of relativity asserts laws and constants same in all inertial frames.
The magnitude of cross product |a×b| equals __________.
A |a||b|sinθ
B |a||b|cosθ
C |a|+|b|
D dot product
Cross product magnitude involves sine of angle.
The conversion factor between radians and degrees is __________.
A 180/π degrees per radian
B π/180 degrees per radian
C 360 degrees per radian
D none
1 rad = 180/π degrees.
For a measurement with significant-digit rule, leading zeros are __________.
A not significant
B significant
C ambiguous
D always counted as one
Leading zeros only place decimal, not significant.
The continuity equation ∂ρ/∂t + ∇·(ρv) = 0 describes conservation of __________ in fluid mechanics.
A mass
B momentum
C energy
D charge
That continuity is mass conservation for fluid with density ρ and velocity v.
The unit of angular velocity is __________.
A rad/s
B m/s
C s
D kg
Angular velocity in radians per second.
In experiment, increasing number of trials improves estimate of mean because of __________.
A law of large numbers
B gravity
C Poisson errors only
D calibration changes
Law of large numbers reduces uncertainty of mean.
The primitive (antiderivative) of 1/r^2 with respect to r is __________.
A −1/r + C
B 1/r + C
C ln r + C
D r^−3/2
∫ r^−2 dr = −r^−1 + C.
A scalar that remains same under rotation but changes sign under parity is called __________.
A pseudoscalar
B vector
C tensor
D scalar
Pseudoscalar changes sign under parity but is rotationally invariant.
The determinant of a reflection matrix in 3D is __________.
A +1 for proper rotation, −1 for improper (reflection)
B always +1
C always 0
D always negative infinite
Improper transformations like reflections have determinant −1.
The most basic coordinate-independent measure of area on a unit sphere is __________.
A solid angle in steradians
B length only
C mass only
D temperature
Solid angle quantifies area on unit sphere.
The Poynting vector in electromagnetism has units of __________.
A power per unit area (W/m^2)
B electric charge
C momentum only
D field potential
Poynting vector represents energy flux (W/m^2).
Which of the following expresses conservation of angular momentum in absence of external torque?
A dL/dt = 0
B dL/dt = τ_external
C L increases always
D None
No torque → time derivative of L is zero.
A measurement with systematic and random errors: which step helps identify systematic error?
A Compare measured mean with known standard/calibration
B Increase number of repeats only
C Reduce instrument size
D Randomize trials only
Calibration against standard reveals systematic bias.
The LHS and RHS of an equation must have same __________ for dimensional consistency.
A dimensions
B numerical values always
C physical meaning only sometimes
D nothing
Both sides must match in dimensions.
If a quantity scales as r^−2, then doubling r changes quantity by factor __________.
A 1/4
B 2
C 4
D 0
r^−2 scaling: doubling r → (2r)^−2 = 1/4 of original.
The quantity cosθ in dot product is __________.
A dimensionless
B has units of radians
C has units of degrees
D has same units as vectors
Cosine of angle is pure number, dimensionless.
The physical quantity with dimension [M][L^2][T^-3] corresponds to __________.
A power (W = J/s) with J = M L^2 T^-2 → divide by T → M L^2 T^-3
B energy
C force
D pressure
Power dimensions are M L^2 T^-3.
Which of the following is true about steradian?
A It is the ratio of area of a spherical patch to r^2 and is dimensionless
B It has dimensions L^2
C It is measured in meters
D It is temperature dependent
sr = area/r^2 dimensionless.
The unit vector êr in spherical coordinates points __________.
A radially outward from origin
B tangent to sphere
C backward
D none
êr is radial outward unit vector.
In coordinate transformations, the volume element transforms by determinant of Jacobian which must be __________ for orientation preserving maps.
A positive
B negative
C zero
D infinite
Orientation-preserving maps have positive Jacobian.
Which of the following is true about conservation laws in physics?
A They often stem from symmetries via Noether’s theorem (for continuous symmetries)
B They are empirical only with no theoretical basis
C They always fail at macroscale
D They only apply to electric phenomena
Noether’s link between continuous symmetries and conservation laws.
The solid angle subtended by a small flat disk of radius a at distance r along axis (a<
A small-angle approximation (a<
B disk is large
C r=0
D none
Small disk area projects nearly perpendicular; approximation valid when a<
A dimensionally correct formula can still be numerically wrong by factor such as 2, π, etc. Example: comparing centripetal acceleration formulas missing factor. This shows dimensional analysis __________.
A cannot detect dimensionless numerical factors
B finds all mistakes
C always gives exact formula
D unhelpful
Dimensional analysis cannot fix pure dimensionless numerical constants.
Conservation of momentum is violated if __________.