Case Study 1: Force on a Charged Particle in a Magnetic Field
A charged particle, such as an
electron or a proton, moving with a velocity v enters a magnetic field B at
an angle \(\theta\) . The magnetic force F acting on the particle can be
calculated using the formula:
\(\ F = qvB \sin \theta \)
where qqq is the charge of the
particle. The direction of the force can be determined using the right-hand
rule.
Questions:
-
The magnetic force on a charged particle is zero when:
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a) It is at rest
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b) It moves parallel to the magnetic field
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c) It moves perpendicular to the magnetic field
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d) None of the above
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If the charge of the particle is doubled, the magnetic
force will:
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a) Remain the same
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b) Double
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c) Triple
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d) Quadruple
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What is the angle that results in maximum magnetic
force on a charged particle?
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a) 0°
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b) 30°
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c) 90°
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d) 180°
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The right-hand rule helps determine:
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a) The direction of velocity
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b) The direction of magnetic force
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c) The direction of magnetic field
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d) The magnitude of force
Answers:
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b) It moves parallel to the magnetic field
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b) Double
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c) 90°
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b) The direction of magnetic force
Case Study 2: Motion of a Charged Particle in a Magnetic Field
A charged particle enters a
uniform magnetic field perpendicular to its velocity. The particle experiences a
magnetic force and undergoes circular motion due to this force. The radius of
the circular path rrr can be determined using the formula:
\(\ r = \frac{mv}{qB}\)
where mmm is the mass of the
particle, v is its velocity, q is the charge, and B is the magnetic field
strength.
Questions:
-
In a uniform magnetic field, a charged particle moving
in a circular path indicates:
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a) The force is constant in magnitude and direction
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b) The magnetic field is increasing
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c) The speed of the particle is changing
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d) The mass of the particle is variable
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If the magnetic field strength is increased while
keeping the velocity constant, the radius of the circular path will:
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a) Increase
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b) Decrease
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c) Remain the same
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d) Become zero
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The frequency of the circular motion of the charged
particle is given by:
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a) \(\ f = \frac{qB}{2\pi m}\)
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b) \(\ f = \frac{mv}{qB}\)
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c) \(\ f = \frac{B}{q} \)
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d) \(\ f = \frac{q}{m}\)
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The centripetal force required for circular motion in
this scenario is provided by:
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a) Gravitational force
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b) Electric force
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c) Magnetic force
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d) Frictional force
Answers:
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a) The force is constant in magnitude and direction
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b) Decrease
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a) \(\ f = \frac{qB}{2\pi m}\)
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c) Magnetic force
Case Study 3: Motion of Charged Particles in Electric and Magnetic Fields
When a charged particle moves
through a region where both electric field E and magnetic field B are
present, it experiences forces from both fields. The net force F on the
particle can be expressed as:
\(\ F = qE + qvB \sin \theta\)
where \(\theta\) is the angle
between the velocity and the magnetic field.
Questions:
-
If the electric field is directed opposite to the
magnetic force, the charged particle will:
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a) Accelerate
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b) Move at constant velocity
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c) Come to rest
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d) Change direction
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The condition for a charged particle to move
undeflected in both electric and magnetic fields is:
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a)\(\ E=B \)
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b)\(\ E=vB \)
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c) \(\ F_E = 0\)
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d) \(\ F_B = 0 \)
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What happens to the particle's velocity when the
electric field is increased?
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a) Decreases
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b) Increases
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c) Remains the same
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d) Becomes zero
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The angle \(\theta \) affects the magnitude of the
magnetic force. What is the maximum value of the magnetic force?
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a) \(\ qvB \)
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b) 0
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c)\(\ qB \)
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d)\(\ qv \)
Answers:
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b) Move at constant velocity
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b)\(\ E=vB \)
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b) Increases
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a) \(\ qvB \)
Case Study 4: Biot-Savart Law
The Biot-Savart Law describes
the magnetic field B generated by a current-carrying conductor. The law states
that the magnetic field at a point in space is directly proportional to the
current I and inversely proportional to the square of the distance rrr from
the wire:
\(\ B = \frac{\mu_0
I}{4\pi r^2}\)
where μ0\mu_0μ0 is the
permeability of free space.
Questions:
-
The magnetic field around a straight current-carrying
conductor is:
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a) Uniform
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b) Radial
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c) Circular
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d) Linear
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According to the Biot-Savart Law, if the distance from
the wire is halved, the magnetic field will:
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a) Halve
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b) Double
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c) Quadruple
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d) Remain the same
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If the current in a wire is increased, the magnetic
field strength will:
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a) Decrease
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b) Increase
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c) Remain the same
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d) Become zero
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The direction of the magnetic field around a straight
conductor can be determined using:
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a) Right-hand rule
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b) Left-hand rule
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c) Ampere's law
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d) Faraday's law
Answers:
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c) Circular
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c) Quadruple
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b) Increase
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a) Right-hand rule
Case Study 5: Ampere's Circuital Law
Ampere's Circuital Law relates
the integrated magnetic field around a closed loop to the electric current
passing through the loop. The law is mathematically expressed as:
∮
\(\ \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \)
where \(\ I_{enc}\) is the
current enclosed by the path.
Questions:
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Ampere's Circuital Law is useful for finding:
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a) The electric field
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b) The magnetic field
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c) The resistance
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d) The capacitance
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If no current is enclosed by the loop, the line
integral of the magnetic field is:
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a) Zero
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b) Positive
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c) Negative
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d) Undefined
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The permeability of free space μ0\mu_0μ0 has a value
of approximately:
-
a) \(\ 4\pi \times 10^{-7} \, \text{T
m/A}\)
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b) \(\ 8.85 \times 10^{-12} \,
\text{F/m}\)
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c) \(\ 9 \times 10^9 \, \text{Nm}^2/\text{C} \)
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d) \(\ 1.6 \times 10^{-19} \, \text{C}\)
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Ampere's Circuital Law can be used to calculate the
magnetic field inside a:
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a) Straight conductor
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b) Solenoid
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c) Capacitor
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d) Resistor
Answers:
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b) The magnetic field
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a) Zero
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a) \(\ 4\pi \times 10^{-7} \, \text{Tm/A}\)
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b) Solenoid