CUET UG Physics Booster Test 2-Ampere’s Circuital Law

Ampere's circuital law is expressed as ∮B⋅dl=μ_0I_(enc) where the line integral is taken over a closed path. The dot product involves the tangential component of the magnetic field along the infinitesimal path element dl. The integral can be visualized as the limit of a summation of many small magnetic field contributions around the loop. The direction in which the boundary is traversed is important because it determines the positive direction of the enclosed current through the right-hand rule. Reversing the traversal direction reverses the sign convention. Therefore, the statement that the sign convention is independent of traversal direction is incorrect.

Ampere's circuital law states ∮B⋅dl=μ_0I_(enc) The algebraic sum of enclosed currents is I_(enc)=2+3-1=4A Substituting into Ampere's law, ∮B⋅dl=(4π×10^(-7))(4)=16π×10^(-7)  T m The sign of each current depends on the right-hand rule associated with the chosen direction of traversal. Since the first two currents are positive and the third is negative, the net enclosed current is 4 A. Therefore, the value of the line integral is 16π×10^(-7) T m.

Ampere's circuital law relates the line integral of the magnetic field around a closed loop to the total current passing through any open surface bounded by that loop. The choice of surface is arbitrary as long as its boundary remains the selected Amperian loop. The surface need not be planar, perpendicular to the conductor or enclose the entire conductor. What matters is the net current crossing the surface. This flexibility is similar to the freedom available in choosing Gaussian surfaces in electrostatics. NCERT emphasizes that Ampere's law depends only on the boundary and enclosed current, not on the exact shape of the surface.

The right-hand rule provides the sign convention used in Ampere's circuital law. When the fingers of the right hand curl in the direction in which the Amperian loop is traversed, the extended thumb points in the direction that is considered positive for the enclosed current. Thus, the curl of the fingers corresponds to the sense of traversal of the boundary loop, while the thumb represents the positive current direction. If the thumb direction is reversed, the positive direction of enclosed current is reversed. Similarly, if the finger curl direction is reversed, the sense of traversal of the boundary loop also reverses. This convention ensures consistency between the line integral (\oint \mathbf{B}\cdot d\mathbf{l}) and the algebraic sign of the enclosed current. NCERT uses this right-hand rule to establish the relationship between loop orientation and current sign in Ampere's circuital law.

An ideal Amperian loop is selected so that the evaluation of the line integral becomes simple. If the magnetic field is tangential to the loop and has constant magnitude, the integral reduces to B×L. If the magnetic field is normal to the path element, the angle between B and dl is 90^∘, making the contribution zero. Similarly, sections where the magnetic field vanishes contribute nothing to the integral. These conditions are frequently used in NCERT derivations involving straight conductors and solenoids. However, the magnetic field need not be parallel to the current element. The field direction is determined by symmetry and geometry, not by the direction of current elements themselves.

Ampere's circuital law states ∮B⋅dl=μ_0I_e When the magnetic field is tangential and has the same magnitude at every point on the Amperian loop, B remains constant throughout the integration. Therefore, ∮B⋅dl=B∮dl Since the total length of the loop is L, BL=μ_0I_e Rearranging, B=μ_0I_e/L This is one of the most important simplifications used in Ampere's law. By selecting a symmetric loop where the magnetic field is constant and tangential, the integral becomes easy to evaluate. This approach is used extensively in NCERT while deriving magnetic fields due to straight conductors, solenoids and toroids.

For an infinitely long straight conductor carrying current I, the magnetic field is given by B=μ_0I/2πr This equation shows that the magnetic field is inversely proportional to the radial distance r from the conductor. As the distance from the wire increases, the magnetic field decreases. Due to cylindrical symmetry, every point located at the same radial distance from the wire experiences the same magnetic field magnitude. Therefore, the loci of constant magnetic field magnitude are concentric circles centered on the conductor. These circular field lines are a direct consequence of the right-hand thumb rule and Ampere's circuital law. Hence, the correct combination is inverse dependence on distance and concentric circles as lines of constant magnitude.

Outside a long straight conductor carrying current I, Ampere's circuital law gives B=μ_0I/2πr This relation shows that the magnetic field is inversely proportional to the distance from the conductor. Therefore, Statement 1 is correct. The derivation is performed using a circular Amperian loop centered on the wire because the magnetic field has cylindrical symmetry and remains constant at every point on the circle. Hence, Statement 3 is also correct. Statement 2 is incorrect because the dependence is inverse first power (1/r) and not inverse square (1/r^2). Statement 4 is also incorrect because the field decreases rather than increases with distance.

For an infinitely long straight conductor, the magnetic field possesses cylindrical symmetry. This means the field magnitude depends only on the radial distance from the conductor and is independent of angular position and axial position. Therefore, Statement 2 is correct. Because of this symmetry, a circular Amperian loop can be chosen such that the magnetic field remains constant over the loop. This greatly simplifies the evaluation of Ampere's integral, making Statement 3 correct. Statement 1 is incorrect because the field does not depend on all three spatial coordinates. Statement 4 is also incorrect because the magnetic field direction is tangential to circular field lines rather than radial.

The magnetic field produced by a long straight current-carrying conductor consists of concentric circular field lines centered on the wire. Unlike electric field lines, which originate from positive charges and terminate on negative charges, magnetic field lines always form continuous closed loops. This reflects the absence of isolated magnetic monopoles. The circular nature of these field lines can be determined using the right-hand thumb rule. Every magnetic field line forms a closed curve and does not have a starting or ending point. NCERT emphasizes that magnetic field lines are fundamentally different from electric field lines in this respect. Therefore, the correct statement is that magnetic field lines form closed loops as concentric circles.

The right-hand rule is applied differently depending on the geometry of the current-carrying conductor. For a long straight wire, the thumb points in the direction of current while the curled fingers indicate the direction of magnetic field lines around the conductor. For a circular current loop, the fingers curl in the direction of current and the thumb points along the magnetic field axis through the center of the loop. A solenoid can be regarded as a collection of circular current loops, so the thumb again indicates the magnetic field direction along the axis of the solenoid. Around a straight wire, the magnetic field forms concentric circles centered on the conductor. These applications are extensively used in NCERT to determine magnetic field directions without mathematical calculations.

According to the right-hand thumb rule, the thumb is aligned in the direction of conventional current while the curled fingers indicate the direction of the magnetic field lines around the conductor. If the current flows vertically downward, point the right-hand thumb downward. The natural curling of the fingers then indicates the magnetic field direction around the wire. When viewed from above, this curl appears clockwise. This result can also be verified by reversing the usual case of upward current, which produces an anticlockwise magnetic field when viewed from above. Therefore, downward current corresponds to clockwise circulation of magnetic field lines.

Inside a solid conductor carrying uniformly distributed current, the magnetic field is given by B=μ_0Ir/2πa^2 Thus, the magnetic field is directly proportional to the radial distance r. At r=a/2, B_1=μ_0I(a/2)/2πa^2 At r=a, B_2=μ_0Ia/2πa^2 Taking the ratio, B_1/B_2=a/2/a=1/2 Therefore, B_1:B_2=1:2 This result directly follows from the linear variation of magnetic field inside a uniformly current-carrying conductor.

For a uniformly current-carrying solid conductor, Ampere's law gives two different field expressions. For r<a, B=μ_0Ir/2πa^2 which shows that the magnetic field increases linearly with radial distance. At the center, the field is zero and it gradually increases toward the surface. For r>a, B=μ_0I/2πr which shows that the magnetic field decreases inversely with distance from the wire. Thus, the magnetic field reaches its maximum value at the surface and then decreases outside the conductor. Therefore, the correct description is "increases linearly" inside and "decreases inversely" outside.

A long solenoid is a tightly wound helical coil whose length is much greater than its radius. This geometry allows the magnetic field inside the solenoid to become nearly uniform. Each turn of the solenoid behaves like a circular current loop and produces its own magnetic field. According to the principle of superposition, the resultant magnetic field inside the solenoid is the vector sum of the magnetic fields produced by all individual turns. Since neighbouring turns are closely spaced, their fields reinforce each other in the interior region, creating a strong and nearly uniform magnetic field. Therefore, all four statements correctly describe the structure and magnetic behavior of a long solenoid.

A finite solenoid is constructed by winding a long insulated wire in the form of a closely packed helix. The insulation is usually provided by an enamel coating on the wire. The purpose of the enamel coating is to prevent electrical contact between neighbouring turns, thereby avoiding short circuits. It does not electrically connect adjacent turns. The magnetic field inside the solenoid is strong due to the combined contribution of all turns, whereas the field outside is comparatively weak. Between neighbouring turns, transverse field components largely cancel because of symmetry and superposition. Therefore, the statement that enamelled wires are used to ensure electrical connection between adjacent turns is incorrect.

To derive the magnetic field inside a long solenoid using Ampere's circuital law, NCERT considers a rectangular Amperian loop abcd. One side of the rectangle lies inside the solenoid parallel to its axis, while another side lies outside where the magnetic field is nearly zero. The transverse sides are perpendicular to the magnetic field direction. Since B is perpendicular to dl along these transverse sections, their contribution to the line integral is zero. The external side contributes negligibly because the external magnetic field is nearly zero for a long solenoid. Consequently, the only significant contribution comes from the segment inside the solenoid that is parallel to the axis. Thus, Statements 1, 3 and 4 are correct.

As the length of a solenoid increases relative to its radius, the magnetic field lines become increasingly confined within the interior region. The magnetic fields produced by different turns cancel each other more effectively outside the solenoid. Consequently, the external magnetic field becomes weaker and tends toward zero. In the ideal case of an infinitely long solenoid, the external magnetic field is considered zero, while the internal field remains strong and uniform. This property greatly simplifies the application of Ampere's circuital law and is one of the key assumptions used in NCERT derivations. Therefore, the correct statement is that the external magnetic field approaches zero as the solenoid becomes longer.

Ampere's circuital law and Gauss's law possess a similar mathematical structure. Ampere's law relates the line integral of the magnetic field along a closed boundary to the current enclosed within that boundary. Similarly, Gauss's law relates the electric flux through a closed boundary surface to the electric charge enclosed within it. In both cases, a measurable quantity defined on the boundary depends only on the total source present in the interior region. This elegant relationship simplifies the analysis of highly symmetric systems and forms one of the central ideas of field theory. Therefore, the correct statement is that both laws relate a quantity on the boundary to a source located in the interior.

Ampere's circuital law and the Biot–Savart law are two complementary methods used to determine magnetic fields generated by steady currents. Ampere's law can be viewed as an elegant alternative expression of the Biot–Savart law, particularly for systems possessing high symmetry. Both laws describe the magnetic consequences of steady electrical currents. Ampere's law specifically relates a boundary quantity, namely the circulation of the magnetic field, to the enclosed current within the interior region. However, the standard form of Ampere's law discussed in NCERT is valid for steady currents and not for rapidly varying currents. The latter situation requires Maxwell's correction involving displacement current. Therefore, Statements 1, 3 and 4 are correct, while Statement 2 is incorrect.