CUET UG Physics Booster Test 3-Ampere’s Circuital Law

Ampere's circuital law states ∮B⋅dl=μ_0I_e To determine the enclosed current, rearrange the equation: I_e=∮B⋅dl/μ_0 Substituting the given values, I_e=12.56×10^(-6)/4π×10^(-7) Using π=3.14, I_e=12.56×10^(-6)/12.56×10^(-7)I_e=10A Ampere's law directly relates the circulation of the magnetic field around a closed path to the total current enclosed by that path. The result depends only on the net enclosed current and not on the shape of the loop. Therefore, the enclosed current is 10 A.

Ampere's circuital law depends only on the net current enclosed by the chosen Amperian loop. Currents that pass outside the loop do not contribute to the enclosed current and therefore do not affect the value of the line integral. When multiple currents pass through the surface bounded by the loop, currents in opposite directions are assigned opposite signs and are algebraically added. The right-hand rule is used to determine which current direction is positive. For a thick wire carrying uniformly distributed current, the current enclosed within radius r is proportional to the area enclosed: I_(enc)=I(r^2/a^2) Thus Statements A, B and C are correct. The incorrect statement is that currents outside the loop contribute to the enclosed current.

The open surface associated with Ampere's law is often visualized as a soap film stretched across a wire loop. The chosen Amperian loop forms the boundary of this surface. The exact shape of the surface may be changed without altering the boundary loop, and the net enclosed current remains the same. Ampere's circuital law involves evaluating the magnetic field line integral around the boundary loop, not over the entire surface area. The surface is used only to determine which currents are enclosed. Therefore, Statements 1, 2 and 4 are correct, while Statement 3 is incorrect because the integration is performed along the closed loop rather than over the surface.

Ampere's circuital law is valid for any closed loop and does not require the loop to be circular. Circular loops are chosen only when symmetry simplifies calculations. The line integral is always evaluated along the closed boundary C of the chosen surface. The surface itself may be distorted into different shapes while maintaining the same boundary. The right-hand rule establishes the connection between the direction of traversal of the loop and the positive normal to the surface. This sign convention ensures consistency when determining the algebraic value of enclosed current. Therefore, Statements 2, 3 and 4 are correct, whereas Statement 1 is incorrect.

Ampere's circuital law is valid for any closed loop regardless of its shape. However, the law becomes especially useful when the current distribution possesses a high degree of symmetry. In systems such as infinitely long straight conductors, long solenoids and toroids, symmetry allows the magnetic field to be constant over selected portions of the Amperian loop. As a result, the line integral simplifies significantly and the magnetic field can be calculated directly. Without symmetry, Ampere's law remains valid but may not provide an easy method for evaluating the magnetic field. In such situations, the Biot–Savart law is often more convenient. Therefore, a high degree of symmetry is the key condition that makes Ampere's law practically useful.

Ampere's line integral depends on the relative orientation of the magnetic field B and the infinitesimal path element dl. When the magnetic field is tangential and constant throughout the loop, the line integral simplifies to BL. When the magnetic field is perpendicular to every element of the path, the dot product becomes zero because cos⁡90^∘=0. Similarly, when the angle between B and dl is zero, the dot product becomes Bdl. These simplifications are extensively used in NCERT while applying Ampere's circuital law to highly symmetric systems. The correct matching therefore relates the integral expressions directly to the geometric conditions that produce them.

An infinitely long straight current-carrying conductor exhibits cylindrical symmetry. Every point located at the same radial distance r from the wire experiences the same magnetic field magnitude. Therefore, all points on a circular path centered on the conductor have identical magnetic field magnitude. According to the right-hand thumb rule, the magnetic field lines form concentric circles around the wire. Consequently, the magnetic field direction at each point is tangential to the circular path. This property allows a circular Amperian loop to be chosen while applying Ampere's law, making the magnetic field constant throughout the loop and greatly simplifying calculations.

The magnetic field due to a long straight conductor carrying current I is B=μ_0I/2πr This equation shows that the magnetic field is inversely proportional to the radial distance r. As the observation point moves closer to the conductor, the denominator becomes smaller and the magnetic field increases. In the ideal mathematical expression, the magnetic field tends toward infinity as r approaches zero. This behavior illustrates the inverse dependence of magnetic field on distance. Although a real conductor has finite radius and the expression is modified inside the conductor, the standard outside-field equation clearly demonstrates inverse proportionality.

For an infinitely long straight current-carrying conductor, cylindrical symmetry implies that the magnetic field magnitude depends only on the radial distance from the conductor. Thus, dependence on angular and axial coordinates disappears, reducing the problem to a single coordinate r. This symmetry allows concentric circular Amperian loops to be selected because the magnetic field remains constant on such loops. Furthermore, the field magnitude is independent of the polar angle around the axis. However, the field direction is not radial; it is tangential to the concentric circular field lines. Therefore, Statements 1, 2 and 4 are correct while Statement 3 is incorrect.

For a circular magnetic field line around a straight conductor, the magnetic field is tangential and constant throughout the circle. Therefore, ∮B⋅dl=B(2πr) Given ∮B⋅dl=4π×10^(-6)  T m and r=10 cm=0.1 m Substituting, 4π×10^(-6)=B(2π×0.1)4π×10^(-6)=0.2πBB=4π×10^(-6)/0.2πB=20×10^(-6)B=2.0×10^(-5) T Hence, the magnetic field magnitude at every point on the circular field line is 2.0×10^(-5) T.

For a long straight current-carrying conductor, the right-hand thumb rule states that the thumb points in the direction of current while the curled fingers indicate the direction of the magnetic field lines around the conductor. In Ampere's circuital law, the right-hand rule is also used to establish the sign convention between the traversal direction of the boundary loop and the positive direction of enclosed current. The fingers curl in the direction of traversal of the loop, while the thumb points along the positive current direction through the enclosed surface. Therefore, Statements A, B and D correctly describe applications of the right-hand rule. Statement C is incorrect because, for a straight wire, the thumb never points in the direction of the magnetic field; it points in the direction of current.

The right-hand rule used in Ampere's circuital law establishes the sign convention between the chosen traversal direction of the boundary loop and the enclosed current. If the fingers curl in the direction of traversal of the loop, the thumb points in the direction considered positive for the enclosed current. Any current flowing opposite to the thumb direction is assigned a negative sign in the enclosed current calculation. This convention ensures consistency between the line integral and the algebraic sign of the enclosed current. However, the rule does not independently define the magnetic field orientation without reference to the chosen loop and surface. Therefore, Statements 1, 2 and 3 are correct, while Statement 4 is incorrect.

For a uniformly current-carrying solid conductor, Ampere's law is applied using a circular Amperian loop of radius r, where r<a. Since current density is uniform, the enclosed current is proportional to the enclosed area: I_(enc)=I(πr^2/πa^2)I_(enc)=I(r^2/a^2) Applying Ampere's circuital law, B(2πr)=μ_0I(r^2/a^2) which gives B=μ_0Ir/2πa^2 Thus, Statements 1, 3 and 4 are correct. Statement 2 is incorrect because the entire current I is enclosed only at the surface r=a, not for points inside the conductor.

For a uniformly current-carrying thick conductor, the magnetic field inside the conductor is B=μ_0Ir/2πa^2 This equation shows that the magnetic field is directly proportional to the radial distance r. Therefore, when B is plotted against r, the graph is a straight line. At the center of the conductor (r=0), the magnetic field is zero because no current is enclosed within an infinitesimally small Amperian loop. As r increases, the enclosed current increases proportionally to the enclosed area, causing the magnetic field to increase linearly. Thus, the graph is a straight line passing through the origin.

A solenoid is considered long when its length is much greater than its radius. Under this condition, edge effects become negligible and the magnetic field inside the solenoid becomes nearly uniform throughout most of its interior. At the same time, the magnetic field outside the solenoid becomes very weak and is often approximated as zero. This assumption is essential in deriving the standard expression B=μ_0nI for the magnetic field inside a long solenoid. The definition depends on geometry rather than current magnitude or turn spacing. Therefore, a solenoid is regarded as long when its length is significantly larger than its radius.

A solenoid consists of a long insulated wire wound in the form of a closely packed helix. Closely spaced turns allow each turn to be treated approximately as a circular current loop whose magnetic field contributes to the overall field. The enamel coating on the wire prevents electrical contact between neighbouring turns and avoids short-circuiting. When the solenoid length is large compared to its radius, the magnetic field inside becomes nearly uniform. A large number of turns causes the magnetic fields of individual loops to add through superposition, producing a stronger resultant field. These structural features are essential for the operation of practical solenoids and are emphasized in NCERT while discussing magnetic field generation by current-carrying coils.

For a long solenoid, the magnetic field inside is given by B=μ_0nI where n=N/L is the number of turns per unit length. Given: N=500,L=0.5 mn=500/0.5=1000 m^(-1) Substituting into the field equation, B=(4π×10^(-7))(1000)(5)B=20π×10^(-4)B=6.28×10^(-3) T Thus, the magnetic field deep inside the solenoid is 6.28×10^(-3) T. This result follows directly from Ampere's circuital law applied to a long solenoid.

To derive the magnetic field inside a long solenoid using Ampere's law, a rectangular Amperian loop is chosen. One side lies inside the solenoid parallel to the magnetic field, while another side lies outside where the magnetic field is approximately zero. Therefore, the contribution from the outer edge cd is taken as zero. The transverse edges bc and ad are perpendicular to the magnetic field direction. Since B⋅dl=Bdlcos⁡90^∘ their contributions are also zero. As a result, only the segment inside the solenoid contributes significantly to the line integral. This simplification leads directly to the expression B=μ_0nI.

Ampere's circuital law relates the circulation of the magnetic field around a closed boundary loop to the current enclosed by that loop. Gauss's law relates the electric flux through a closed surface to the enclosed electric charge. Thus, both laws connect a quantity defined on a boundary to a source located within the enclosed region. Ampere's law specifically employs a line integral along a one-dimensional closed path. Both laws are particularly useful for highly symmetric systems such as infinite wires, solenoids, spherical charge distributions and cylindrical charge distributions. Statement 3 is incorrect because the standard NCERT forms are not restricted exclusively to rapidly varying fields.

Oersted's experiment demonstrated that an electric current produces a magnetic field. Ampere's circuital law provides a quantitative and theoretical framework for understanding this observation by relating magnetic field circulation to enclosed current. The law is especially useful for continuous current distributions possessing high symmetry, such as infinitely long conductors and long solenoids, where direct magnetic field evaluation becomes straightforward. Statement 2 is incorrect because Ampere's law can be derived from the Biot–Savart law under steady-current conditions. Statement 3 is also incorrect because electric and magnetic fields are distinct physical entities, and magnetic monopoles have not been experimentally observed. Therefore, only Statements 1 and 4 are correct.