NYJC EJC 2026 Electric and Magnetic Fields Tutorial

9814 H3 Physics (2026) 1 Electric and Magnetic Fields tutorial 1 The six faces of a cubical box each measure 20.0 cm by 20.0 cm, and the faces are numbered such that faces 1 and 6 are opposite to each other, as are faces 2 and 5, and faces 3 and 4. The flux through each face is given in the table. Determine the net charge inside the cube. face flux / N m2 C−1 1 −70.0 2 −300.0 3 −300.0 4 300.0 5 −400.0 6 −500.0 2 Electric fields of varying magnitudes are directed either inward or outward at right angles on the faces of a cube, as shown in Fig. 2.1. Fig. 2.1 Determine the strength and direction of the field on the face F. 3 A −6.00 nC point charge is located at the centr e of a conducting spherical shell. The shell has an inner radius of 2.00 m, an outer radius of 4.00 m, and a charge of 7.00 nC. (a) Determine (i) the electric field at r = 1.00 m, (ii) the electric field at r = 3.00 m, (iii) the electric field at r = 5.00 m. (b) Calculate the surface charge distribution, , on the outside surface of the shell. 4 A hollow conducting spherical shell has an inner radius of 8.00 cm and an outer radius of 10.0 cm. The electric field at the inner surface of the shell has a magnitude of 80.0 N C −1 and points toward the centre of the sphere, and the electric field at the outer surface has a magnitude of 80.0 N C−1 and points away from the centre of the sphere. Determine the charge on the inner surface and on the outer surface of the spherical shell. 5 Two parallel, infinite, non-conducting plates are 10.0 cm apart and have charge distributions of 1.00 C m−2 and –1.00 C m−2. Calculate the force on an electron (a) in the space between the plates,

9814 H3 Physics (2026) 2 (b) located outside the two plates near the surface of one of the two plates. 6 Two parallel, uniformly charged, infinitely long wires are 6.00 cm apart and carry opposite charges with a linear charge density of 1.00 C m−1. Determine the magnitude and direction of the electric field at a point midway between the two wires and 40.0 cm above the pla ne containing them. 7 A thin, hollow, metal cylinder of radius R has a surface charge distribution . A long, thin wire with a linear charge density 2  runs through the centre of the cylinder. Determine an expression for the electric field and determine the direction of the field at each of the following locations: (a) rR (b) rR 8 Two infinite sheets of charge are separated b y 10.0 cm as shown in Fig. 9.1 . Sheet 1 has a surface charge distribution of 2 1 3.00 C m −= and sheet 2 has a surface charge distribution of 2 1 5.00 C m −=− . Determine the total electric field at (a) P, 6.00 cm to the left of sheet 1, (b) P’, 6.00 cm to the right of sheet 1. Fig. 9.1 9 A non-conducting sphere has a uniform volume charge density . Let r be the vector from the centre of the sphere to a general point P within the sphere. (a) Show that the electric field at P is given by    03 r . (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 11.1. Fig. 11.1

9814 H3 Physics (2026) 3 Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to    03 a , where a is the position vector from the centre of the sphere to the centre of the cavity. (Note that this result is independent of the radius of the sphere and the radius of the cavity) 10 Suppose the charge density of a solid sphere in is given by 2r= where  is a constant. (a) Determine  in terms of the total charge Q on the sphere and its radius 0r . (b) Determine the electric field as a function of r inside the sphere. 11 State a similarity and two differences between the application of Gauss’s and Ampere’s laws. 12 The current I within a wire that has a circular cross section of radius R is known to be distributed uniformly over that cross section. (a) Determine the magnetic field as a function of the distance r from the wire's axis where (i) r  R (ii) r  R (b) Sketch the B – r graph. 13 A very long, cylindrical wire has an insulating core surrounded by a cylindrical conductor carrying current I. The current is uniformly distributed in the conducting cylinder. The radius of the insulating core is r1, and the outer radius of the conducting cylinder is r2. Determine expressions for the magnetic flux density (a) outside the conducting cylinder (r  r2), (b) inside the conducting cylinder but outside the insulator (r1  r  r2), (c) and inside the insulator (r  r1).

9814 H3 Physics (2026) 4 14 Show that a uniform magnetic field cannot drop abruptly to zero. 15 A toroid having a square cross section, 5.00 cm on a side, and an inner radius of 15.0 cm has 500 turns and carries a current of magnitude 0.800 A. Determine the magnitude of the magnetic field inside the toroid at (a) the inner radius and (b) the outer radius of the toroid. 16 The current density in a cylindrical conductor of radius R varies as ( ) 0JrJr R= (in the region from zero to R). (a) Express the magnitude of the magnetic field, B, in the regions r  R and r  R. (b) Sketch the graph of B against r. 17 A current loop, carrying an amount of current of 5.0 A, is in the shape of a right triangle with sides 30 cm, 40 cm, and 50 cm. The loop is in a uniform magnetic field of magnitude 80 mT whose direction is parallel to the current in the 50 cm side of the loop. Calculate the magnitude of the (a) magnetic dipole moment of the loop, (b) torque on the loop.

9814 H3 Physics (2026) 5 18 Calculate the magnitude of the force, due to an electric dipole of dipole moment 3.6  10−29 C m, on an electron 25 nm from the centr e of the dipole, along the dipole axis. Assume that this distance is large relative to the dipole’s charge separation. 19 Figure below shows an electric quadrupole. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Determine the magnitude of E  on the axis of the quadrupole at a point P a distance z from its centre. Assume z  d. 20 An electric dipole consists of charges 2e and −2e separated by 0.78 nm. It is in an electric field of strength 3.4  106 N C−1. Calculate the magnitude of the torque on the dipole when the dipole moment is (a) parallel to, (b) perpendicular to, and (c) antiparallel to the electric field. 21 Sketch the graph of (a)  against  (b) U against  of an electric dipole placed in a uniform electri c field, where  is the torque, U is the electric potential energy and  is the angle between the dipole moment and the field. 22 Figure below shows the potential energy U of an electric dipole against the angle  between a uniform electric field E  of magnitude 20 N C−1 and the dipole moment p  . Calculate is the magnitude of p  . 23 Determine the frequency of oscillation of an electric dipole, of dipole moment p  and moment of inertia I, for small amplitudes of oscillation about its equilibrium position in a uniform electric field E  . 24 An electric dipole consists of an electron and a proton at a separation of 2.00 nm and it is in a uniform field of magnitude 3.00  106 N C−1. Calculate the amount of energy required to turn an electric dipole from being lined up with the electric field to being lined up opposite the field.

9814 H3 Physics (2026) 6 25 An infinite line of positive charge lies along the y axis, with charge density  −= 12.00 C m . A dipole is placed with its center along the x axis at x = 25.0 cm. The dipole consists of two charges of magnitude 10.0 C separated by 2.00 cm. The axis of the dipole makes an angle of 35.0  with the x axis, and the positive charge is further from the line of charge than the negative charge.